Bulk-boundary correspondence for disordered free-fermion topological phases
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Max, Christopher
Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C^\ast-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator K-theory (or KR-theory): a bulk class, using Van Daele's picture, along with a boundary class in Kasparov's Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these KR-theory classes to each other.
 In my talk, I would like to present you some results that I recently developed in joint work with Alexander Oledich and Martin Zürmauer at the University of Cologne. The topic is, as the title says, bike boundary correspondence in disorder systems, but primarily it's about how to derive topological phases in disorder systems and how to well motivate them from physical principles. And we are not restricting to single particle picture but to, yeah, free fermions, so no interaction is in its talk, but as we don't consider fixed particle number, we have to work in a more general setup of the number space. I'm not sure if anybody knows, so I give a short reminder at the beginning what the number space picture is for many particle systems. And once I've settled this, then I will go over to topological classification in the bike. The essential assumption there will be that my systems are gapped. And once I've established this classification for all similarity classes of the Eutern-Symbar table, I would then go over and do a similar thing on the boundary where, of course, it's essentially that we have a gapless spectrum. And once I've put this to definitions of topological phases on form mathematical grounds, I can formulate the bike boundary correspondence. Okay, so let's start with the Nambu space picture. The fundamental ingredient of my picture is some single particle Hilbert space. In my case, I will only consider solids. It's the usual A2 space over ZD, and I have some finite dimension in a vector space, which represents a local degrees of freedom. All this is complex Hilbert space, and any given ground state or any chemical potential determinants distinction in the Hilbert space into conduction states and valence states. So and a given ground state at zero temperature is then given by the state where all valence states are filled and all conduction states are empty. Now in the many particle picture, divisions from this ground state are described by the Fox space. So here it is essential that we have here the conduction states and the dual of the valence states. This is because we want that the zero degree in this direct sum represents the ground state, and therefore we have to take the holds as the dual space. So it's not just V what's in there. And to further understand this, the meaning of this, we can split this up as written here, and then the P denotes the number of states over the chemical potential and Q denotes the holds below. So the fundamental physical operations I can do on this Fox space now are either insertion of a state over the potential. So this is given by the usual insertion of a state. Of course we are considering fermions, so we have to anti-symmetrize. Or I can insert a hole below the chemical potential and that's the other. And I can do the reverse and remove a state from above or below. So from these fundamental physical operations I can then define the creation and elation operators that probably everybody knows. So these are the creation operators written in these terms and these are the annihilation operators. So note that the annihilation operator indeed annihilates the ground state. So you can check that our beginning that the Fox space is built on V plus plus V minus dual is indeed the correct space to start. And well, so from this picture we have now the creation and annihilation operators and these define the number space. So the number space is defined as the linear vector space spent by all these operators and it can easily be shown that this space is isomorphic to V plus V dual. So by identifying the creation operators with V and the annihilation operators with V dual. And this is a linear C linear isomorphism. So as you might directly see here this number space comes with the canonical real structure which is always there which is a part of the whole conjugation. So H was the real isomorphism from V to V dual and this can always be defined. Furthermore we have the canonical anti-commutations relations on these creation and annihilation operators. For fermions these read as here where the curly bracket denotes the anti-commutator. And these relations induce via these isomorphism symmetric bilinear form on the number space. So in total I got a symmetric bilinear form and the real structure and of course I can then can clearly define in a product by combining these two informations and the take a message from this slide is basically that the number space is a real Hilbert space just by construction. So here's also essential that we are considering fermions because for bosons we would have to replace the anti-commutator by the commutator but this would lead to minus sign here and here and then we don't have a positive definite inner product so that's no longer Hilbert space. So yes? What is the isomorphism? Is it just the obvious? Yes it's from here you can see that this is linear. The argument so you can map this to V but this is anti-linear in the argument because here you have the first isomorphism so I have to identify with the dual in order to get a linear isomorphism. So usually this one is not canonical. What is not canonical? The resismophism is canonical if you have an inner product on a Hilbert space. Yes I need the inner product on the single particle Hilbert space. Yes? Yes there's. Yes it's an even. Why would you get one to odd dimension? No because you said the non-phosphate is a real dual space so you really must have some even dimension, generalized even dimension within the structure. Yes I mean. So you can't get a situation. So the real dimension is even yes. With the complex dimension. Okay so now operators on this number space so general operators in the magnetic particle picture with interactions would just be contained in the Clifford algebra over W as Chris for example considered. So the CRI algebra of course you have to need to have further conditions like even parity and stuff like this and the Q is just the quadratic form that is induced by the car form. But I want I only can consider non-detecting operators and these are generated by these quadratic order operators and one can easily show that these are isomorphic again by linear isomorphism to the linear operators over my number space. And if I only want to consider single particle operators with fixed particle number then I would only allow these two and these would correspond to really indeed linear operators over my single particle space. So then I would recover the usual picture of single particle physics. Okay so the crucial observation is now that time evolution has to be unitary and orthogonal. I think unitary is well established assumption and orthogonal means here that the unitary time evolution has to preserve the anti-commutation relations between the creation and annihilation operators. So that's also a reasonable assumption and together this means that all Hamiltonians that are physically reasonable have to be self-enjoyed and skewsy metric or post differently they have to be self-enjoyed and imaginary with the real structure that is induced on the operators over the number space. So that's now fixed and this is my starting point for my classification. Now in the classification in the bike as I said before I make the fundamental assumption that the bike is kept. So I can consider always the flattened Hamiltonian that squares to one. I have one question there. Yes? Non-interacting setting needs to be enforced already at that level or? In order to consider operators over the number space it's necessary. Unitary evolution needs to be orthogonal. Well the complex over the space would be double-dimensional. Yes. So I was wondering how you were allowed to speak for generic interaction. No, no interaction. This Hamiltonian is in the sense of Hamiltonian before second quantization. Which one? Right. The one on the previous slide. It's a general Hamiltonian over the number space. Yes. That's the one. That's the one. Still state space is some sort of second quantization of number space. Right. But in real representation that's why you... So this observation you cannot move it. After the first thought you need to have a mainstream... Yeah. I didn't thought about this in detail yet. Okay. So the aims of the next section are now to establish first of all systematic description of this order of solid state systems in this number space picture. And then derive a systematic and constructive. That was one of our central motivations to make everything constructive so that in the end you indeed can calculate stuff and do this for a symmetric crisis of the tenfold way. Okay. So in the bike the single particle space as said before is A2 of ZD. So any bravado letters with some internal space. And the corresponding number space is B plus B dual. So I can write it in this form. And similar to myos talk I introduce finite range operators as written here. So it's just finite hopping of operators. And the C star that is generated by all these finite range operators is the mathematically equal to the uniform row C star as observed by Kubota. And I denote this row C star algebra by this type here. Okay. So in principle one can start with this C star algebra but as we learned yesterday the K theory is not very treatable. So we have to go further. And instead of going to the non-uniform row C star algebra it was one possible way I introduce this order in my systems by using the techniques by Belissar and many authors that are here in the audience. And this goes as follows. I consider a disorder configuration space. For simplicity I assume that it's of this form. So omega naught denotes some kind of disorder on each unit cell. And I copy it for each unit cell. So I got a canonical z, d action by translations on this disorder space and for technical reasons I also need a probability measure. But with this I now consider the so-called disorder observables which are defined as all the functions from my disorder space into the set of allowed operators over my letters such that this covariance condition is satisfied. This covariance condition can be thought of as a tautological condition because the disorder breaks the translation symmetry. So if I consider one fixed disorder configuration and shift it this should be the same as shifting the underlying letters. So this has to be satisfied and thus we make this assumption here. And this will be the set of all disorders observables over the bike letters. Now the use of the translations over the letters. So lambda is the, yeah just, yes exactly. Yeah so I translated everything into linear operators over my number space so that I don't have to consider c and c dagger anymore because mathematically it's the same. So that's why I don't use this here. And the name homogeneous disorder can be understood by this observation here that for any given t in my disorder observable algebra I can average over all disorder configurations and this yields a translation invariant operator. So if I go zoom out very largely out of my solid then I would recover a translation invariant system again. So and no disorder is simulated by setting the local disorder to a point. And then this is just translation invariance. Okay so there's a theory in Barrow from 2003 that if I am in the clean case then this c-style algebra is just the usual c-style algebra of translation invariant operators. So I can consider the torus again and do the usual classification. Now in our setting we have a general, more general theory that if I consider any disorder configuration space then I can show that our c-style algebra is isomorphic to the usual c-style algebra, the cross product c-style algebra that is used by Belisar and Fodontovs Byser and many other authors. And this cross product c-style algebra can be considered as the closure of all polynomials of this form. So the u again denote the shifts along the letters and the m is now a disorder valued local parameter. So by these natural physical constructions I in particular showed that I always can write my hematomines in such a polynomial form. Okay. All right. Now if there are no questions to this I will now go on to explain symmetries to you. Of course I want to consider all symmetry classes of the tenfold ways so I have to explain to you what I mean by symmetries. The symmetry we define it as a real so it preserves the structure of my real hibbert space unitary or anti-unitary. This means anti-linear isomorphism that commutes with the Hermitonian. This differs slightly from other definitions of symmetries but we only allow symmetries to commute with Hamiltonians. And in this talk it is sufficient to consider only local symmetries that only act on the internal space so this w. My physical examples of local symmetries are of course time reversal symmetry. Initially it acts only on the single particle hibbert space but by my construction to the number space it lifts to an anti-linear operator over my number space which is of this form. Then I also consider charge number conservation. This corresponds to the symmetry iq where q measures the number the charge is the charge operator. So q is 1 on particles and minus 1 on holes because they have opposite charge. And I have to insert the i here because I demand that all operators are real or symmetries but of course my real symmetry comes from part of your whole conjugation and charge anti-commutes with part of your whole conjugation because holes have opposite charge than particles. And another one is particle hole transformation so this is an anti-linear operator that almost looks like my particle hibbert conjugation but it is enriched by some unitary which acts on a single particle hibbert space and squares to one. Okay, these are some examples and now in order to perform a classification I could in general try to classify all flattened Hamiltonians in age but from a mathematical point of view these is a bit difficult because the Hamiltonians are imaginary so it's quite unnatural to consider them. Instead it's much more practical to consider the corresponding real operators which we call quasi-particle vacua which is just an i in front of the age in the minus sign of convention. So then of course I have to translate Hamiltonians with these local symmetries into quasi-particle vacua with symmetries and these symmetries are called pseudosymmetries which is introduced by Kennedy and Zambor. And these are defined as follows, a quasi-particle vacuum here the Riemann under what it contains as algebraic conditions has r positive and s negative pseudosymmetries if there's a clifford action into my algebra such that I have r operators that are self-adjoint and square to 1 and s operators that are skew adjoint and square to minus 1 and all of these anti-commute with my j. So the small k and small j denote the real generators of my clifford algebra. Okay, and my claim is that all symmetry classes of the tenfold way can be written, can be represented in this form. So just an QPV that represents the ground state plus a clifford action. And that's what I want to explain now. So let's consider only the real symmetry classes, the complex symmetry classes are already explained in very detail in the book by Proudhan and Schultz-Biles for example so I won't consider them here. So let's start with class D, class D corresponds physically to a superconductor with no symmetries so I also got no pseudosymmetries and that's it. So class D3 corresponds to a superconductor with time reversal symmetry and time reversal symmetry as explained is an antilinear isomorphism on W which commutes with my h but is antilinear therefore it anti-commits with j which is just i times h. So I can construct exactly one pseudosymmetry by this here. Okay, so the next class is A2, A2 is a topology insulator because I have charged conservation with time reversal symmetry. And there I can again define this one but furthermore I can define another one by defining J2 equal i gamma, gamma was the real structure on the number space times t times q and one can verify that this indeed anti-commits with j and squares to minus one and the skew here mission. Okay so the next is a topology insulator with time reversal symmetry and a particular Holtz transformation one may also call it a chiral symmetry and then one can do similar steps and define a third pseudosymmetry and so on and so forth. Maybe class, I also mentioned class C because it's not the usual form on encounters in the literature for the symmetry classes. So class C is a superconductor, so no charge conservation with spin rotation symmetry. Spin rotations are generated by three operators corresponding to the polymetrices but again I have to put an i here in order to make them real and they commute with h if my hematoma is spin symmetric. And then I have to go to two by two matrices in order to define four pseudosymmetries and these are then of this form and my j has this form and yeah. H and minus h yeah. Okay so that's a little bit different but the other cases are then formed in a similar way just by defining the pseudosymmetries that I explained before in this block diagonal form. So and these are all eight classes then translated to QPV with species symmetries. Okay so now after all my physical input is some QPV and some input of symmetries that's all and I have a of course a fixed index of positive and negative species symmetries and now I want to consider the set of all allowed QPV such that this is a symmetric QPV. And I start my classification by considering all homotopy equivalence classes in this set of allowed ground states or QPVs. If I only consider homotopy theory I could stop here and just calculate this set but I want to do k theory and therefore I want to stabilize and in the first step I then explain what I mean by QPVs over matrix algebras over a so if I get more bands in my system then I define it in this way where phi n denotes the direct sum so the diagonal action of phi and then of course I can consider homotopy equivalence classes on this larger algebra. Now in order to compare QPVs in different matrix algebras I have to somehow explain how I do this and this is done by fixing some ground state or QPV which I call topological trivial. I have to fix this and then I can say okay my QPV in my smaller algebra is the same as the QPV in a larger algebra if it's just a direct sum with some topological trivial summand. And this forms an inductive system and then if I go to n to infinity so I allow any degrees of freedom added then I finally obtain a set of stable homotopy equivalence classes and then my set of topological phases is defined as the set of all topological phases I defined in this way but always relative to each other. This means I only can tell you what are topological phases relative to each other. I have no absolute notion of topological phase. This is only fixed by this J-Ref. Okay and then okay maybe I just this is a bit technical I will skip this. The only take home message of this is this physical motivated set of topological phases can be identified with van Dalles K theory where van Dalles K theory is built upon odd self adjoint unitary matrices but since I have not much time I will skip this here and just tell you that I can identify my group of topological phases with K theory via this identification. So in the end I obtained a canonical definition of topological phases for some QPV relative to some reference QPV in this way. And yeah okay so after all I have a new motivation for the usual cross product C star algebra as the algebra of observables and yeah my definition of a topological phase only depends on R and S so maybe on my phi here so it doesn't distinguish between the different symmetry classes. And this is a bit hard to understand at the beginning that I have to fix a trivial topological phase but it's like this and I can identify with K R theory. Alright and as an advertisement with the systematic constructions I also can now define systematic sets of indices. So I obtained that my topological phases live in such a K theory group here and for any given index R, S I can define an index so this is only a Z-valued index by this formula so it's an adapted formula for the complex cases it's a so-called conjourn sorry character. So this is just an advertisement it's not possible to explain it in a way that it's completely understandable but so just yeah. Not yet. Yes. You're fine in K R, it's just how you get a number. Is this pairing only these non-torsion invariants? Yeah currently I'm trying to use Kehl-Endon's results in order to also derive formulas for the torsion values but that's a bit more technical and more difficult. Yeah. This these J's? Yes. So you always insist that they must be from the algebra. Yes. But they could be the automorphism group. And it's not possible in general to directly identify them with K R theory I think. So yeah my assumptions are essentially based on this because yeah. I mean you're constructing the J-1 to J-S from the symmetry operators so I mean they're just in the algebra. I think they're even point wise right so they were just a matrix algebra and then everything is in a any way. Yeah right in this case yes. Not local symmetry where restructure is mixing things at different places. If you really want it. Well I could extend my notions to non-point wise things but in the end I have here a projection so this is a corner algebra and what I need to in order to identify with the K theory of this inner algebra I need to have that the K theories are the same. So for this I'm not sure how this works for outer symmetries. But I think that you need to extend this one on the fox base. Okay so now shortly the top of the classification at the boundary. So I introduce a boundary as done often in the literature by just considering an infinite boundary perpendicular to one translation and direction. So in this way I obtain the half space letters which is just the right hand letters here and the boundary letters is still ZD so it's all respects the translation and directions in the ZD minus one direction and the primary difference to the bike is that my shift operators perpendicular to the boundary are no longer unitaries because of course if I shift something over this boundary I map it to zero. So I obtain this condition here where P0 is the projection on this edge here this boundary. But besides this fact I can now define the algebra absorber with over this half space letters in the same way as in the bike again I have this covariance condition and here I put in the half space letters and the same for the boundary. Yeah there again I don't have a head because all shifts in this direction are still unitary. But basically the definitions are the same and I can show that the half space algebra is now isomorphic to the cross-product C star algebra with the semi-group ZD minus one times N and this can be considered as the closure of all non-commutative polynomials of this form where this pre-factor is now in the boundary algebra. So all shifts are contained in here with a parallel to the boundary. Alright so now what's with the QPVs? Yeah first of all I want to explain to you the canonical morphism from the half space to the bike this is important to distinguish in the end between boundary observables and bike observables in the half space. So and this is done by a bike projection from a half space which just maps the half space translation operators onto the bike translation operators and in this way I obtain a set of boundary observables that are localized at the boundary and these are defined as all operators that are in the kernel of my bike projection. So all observables that are vanish if I go into the bike have to be at the boundary. So and this is the idea that is generated by the boundary projection and this can be shown to be isomorphic to this C star algebra. Okay and as I said here this is the observation physically and I can put this information in a shorted sequence so these are the boundaries this can be identified with some set in the half space and on the right hand side I got the bike and these all together from the half space. I don't think you define what this A delta actually is. A hit? No A delta. I mean I know that you say that it's isomorphic to this kind of thing. This one here. Okay so this is the D-dimensional version of the bike I just showed you. Bike algebra. No. Yes here it's important that the symmetries are on site. Yes. Yes indeed. So the symmetries are on site but the J is not on site so the J cannot be unitary in general anymore. So of course I have gap less boundary states. There's no gap anymore. So what I only can demand is that when I project into the bike then I have a gap operator. That's my definition of a QPV with boundary. So that when I project in the bike I recover a QPV. And then I can follow by general properties of my boundary projection row that J has to be a QPV so it has to be squared to minus one up to boundary states. And this is the operator encoding of gap less boundary states. And the unitary star morphism since it's act locally is still a unitary star morphism now in the half space algebra. But I cannot demand anymore that it's exactly anti commutes with J because yeah this does not have to be satisfied in general from an algebraic point of view. And there's a canonical association between bike QPVs and half space QPVs by a section of this short exact sequence. Now I would like to establish some kind of topological classification. So some kind of homotopy equivalence between QPVs with boundary. But I have in my mind that I want to classify the properties at the boundary. So if I just consider a homotopy of operators in a head this would still contain some informations about the bike because the half space contains both the boundary and the bike. So I have to somehow lose my conditions on my homotopy. And this is possible by going over to the so called multiplier algebra over my localized boundary states. And this can be defined as all operators over the half space. Here I still have the disorder in this picture that preserve the boundary states. So this means that this T is times some operator in this boundary states. This is still a boundary states. In particular these are still covariant with respect to the translations parallel to the boundary. In other words this means that I preserve the covariant conditions parallel to the boundary. And I no longer demand that my operators in this algebra are covariant with respect to the perpendicular translations. So in my opinion this is a reasonable assumption because a boundary destroys translation invariant by construction. So it doesn't make any sense to assume this covariance if I'm interested in properties at the boundary. So that's why we dropped this condition. And furthermore there's an integrity, more stomaphism from my general algebra into the larger algebra. And yeah, so this now generalizes the QPV with boundary to any set of two builds where F is now satisfies the same conditions as J. But the only difference is that it's now an element of the multiplier algebra. And my phi now also maps into the multiplier algebra but I still have the same conditions as before. And I will call such a topological trivial, oh there's a typy here, there should be F's here. Sorry for that. I call such a trivial if all of these conditions are equal to 0 of these quantities. So if I still got exact symmetries my J hat is still skew Hermitian but crucially my J hat squares to minus 1. And as we learned before, if J hat squares to minus 1 this means that there are no gap plus boundary states. So from a physical point of view this is a natural notion of topological trivial phase. And under those by definition this set defines a K group by modding out these topological trivial phases and considering only homotopy equivalence classes of these two builds. So in this way I have a second picture of topological phases which is again a K group but now from a completely different point of view. And to summarize now the boundary topological phase of any disorder topological insulators or superconductor is defined as the equivalence class that is generated in this set of topological phases at the boundary. And some algebraic properties that follow by these constructions are for example that I always can demand that my symmetries are again preserved exactly so I can demand that my symmetries are still anti commuting with J hat and I also can demand that my J hat is skew Hermitian. And so the only fact that prevents topological phase from being trivial is that it's not in general squares to minus one and if it does then it's trivial. And as I said before the section of my boundary Schottx sequence associates to any given bike QPV a canonical boundary QPV with boundary. And to summarize from a physical point of view two boundary topological phases are equal if and only if there's a path in this multiplier algebra such that these are two are equal up to trivial phases. And furthermore a boundary topological phase is trivial if and only if there are no stable stable in the sense of homotopy classes, Gapless boundary states. So yes that's another observation here the reference is already there it's the one where there are no Gapless boundary states. Why? I don't quite get the picture. Yes. But these are two are private two independent half spaces. So. So there is the folding trick where you fold it and then you reuse interface the backroom. Okay. Okay. How much time? Yeah, I can either make a short or I mean I already have explained some to some extent the topological phases and now the bike boundary constant is quite easy to construct because yeah from a mathematical point of view the bike boundary constant is established on the K theory. I have already introduced the bike boundary short exact sequence here as a reminder this is the set of observables that are localized at the boundary half space and the bike and this induces always a long exact sequence on the K theory which is of this form but the only relevant morphism for us is the one that is here diagonal. It's the one that maps bike topological phases onto boundary topological phases and I can interpret it like this because I have explicitly identified my topological phases with K R theory groups and thus I immediately get a bike boundary cosponence by this morphism which is constructed from abstract mathematical concepts. And now the main result of our work is that this bike boundary morphism maps topological phase in the bike which is measured with respect to some J ref onto the boundary topological phase and again here occurs the reference KPV. This might surprise you a bit but if this J ref is for example an atomic limit topological insulator or superconductor then this phase is trivial so then indeed I only get this class here so what one actually should expect but of course if I consider here any possible difference of two KPVs then the difference has to be observed at the boundary too. And these J heads are the canonical lifts from my bike to my half space. Okay so some, finally some physical properties. The image of the bike boundary cross bondence is equal to the set of all boundary topological phases if my omega naught is contractable but this doesn't have to be true in non-node more general settings of this order. Or stated differently all boundary topological phases are corresponding to some bike topological phases so I cannot just change the boundary without changing the bike. So one physical formulation of bike boundary cross bondence. And another point is given a fixed bike class the boundary class depends on the direction of the boundary. Probably most of you will notice but for me it was a bit surprising that you measure different topological phases depending on which phase you cut your system in. It's not all the same. It's only the same for the strong topological invariant. This is the same on all boundaries up to a sign which depends on some orientation of your space but this is seen on all phases and this also applies that if I got a non-trivial strong topological invariant in the bike then all boundaries have gapless localized states which cannot be destroyed by local. Well the bike is gapped so I cannot move any gapless state with small amount of energy into the bike. So the mathematical statement is I have always states in the gap. Right? But they don't specularly localize that. Localized. So that's what you mean right? They're very much stretched along the bike. They're very non-local. Localization is still there. Yeah okay. They're transversely localized. Yeah the mathematical assumption was in the bike I have such a gap spectrum and the statement is if I introduce a boundary then I get some gapless spectrum and since I know it cannot be in the bike it has to be at a boundary. That's how I think about it. Yeah. Yeah? To make a statement that this is what we call the spectral statement and there is a dynamical statement on top of this which is that these gapless boundary states do not understand localize. What we found is for sure you need pairing to go all the way into index theory to resolve the second layer of the problem. Okay. Which is the dynamical statement. Okay. So that's why the word for your harness is pairing is quite important. Okay yeah. But yeah. Are you finished with the work yet? Yeah. So let's give it a hand. Yeah. Thank you.
 In my talk, I would like to present you some results that I recently developed in joint  work with Alexander Oledich and Martin Zürmauer at the University of Cologne.  The topic is, as the title says, bike boundary correspondence in disorder systems, but primarily  it's about how to derive topological phases in disorder systems and how to well motivate  them from physical principles.  And we are not restricting to single particle picture but to, yeah, free fermions, so no  interaction is in its talk, but as we don't consider fixed particle number, we have to  work in a more general setup of the number space.  I'm not sure if anybody knows, so I give a short reminder at the beginning what the number  space picture is for many particle systems.  And once I've settled this, then I will go over to topological classification in the  bike.  The essential assumption there will be that my systems are gapped.  And once I've established this classification for all similarity classes of the Eutern-Symbar  table, I would then go over and do a similar thing on the boundary where, of course, it's  essentially that we have a gapless spectrum.  And once I've put this to definitions of topological phases on form mathematical grounds, I can  formulate the bike boundary correspondence.  Okay, so let's start with the Nambu space picture.  The fundamental ingredient of my picture is some single particle Hilbert space.  In my case, I will only consider solids.  It's the usual A2 space over ZD, and I have some finite dimension in a vector space, which  represents a local degrees of freedom.  All this is complex Hilbert space, and any given ground state or any chemical potential  determinants distinction in the Hilbert space into conduction states and valence states.  So and a given ground state at zero temperature is then given by the state where all valence  states are filled and all conduction states are empty.  Now in the many particle picture, divisions from this ground state are described by the  Fox space.  So here it is essential that we have here the conduction states and the dual of the  valence states.  This is because we want that the zero degree in this direct sum represents the ground state,  and therefore we have to take the holds as the dual space.  So it's not just V what's in there.  And to further understand this, the meaning of this, we can split this up as written here,  and then the P denotes the number of states over the chemical potential and Q denotes  the holds below.  So the fundamental physical operations I can do on this Fox space now are either insertion  of a state over the potential.  So this is given by the usual insertion of a state.  Of course we are considering fermions, so we have to anti-symmetrize.  Or I can insert a hole below the chemical potential and that's the other.  And I can do the reverse and remove a state from above or below.  So from these fundamental physical operations I can then define the creation and elation  operators that probably everybody knows.  So these are the creation operators written in these terms and these are the annihilation  operators.  So note that the annihilation operator indeed annihilates the ground state.  So you can check that our beginning that the Fox space is built on V plus plus V minus  dual is indeed the correct space to start.  And well, so from this picture we have now the creation and annihilation operators and  these define the number space.  So the number space is defined as the linear vector space spent by all these operators  and it can easily be shown that this space is isomorphic to V plus V dual.  So by identifying the creation operators with V and the annihilation operators with V dual.  And this is a linear C linear isomorphism.  So as you might directly see here this number space comes with the canonical real structure  which is always there which is a part of the whole conjugation.  So H was the real isomorphism from V to V dual and this can always be defined.  Furthermore we have the canonical anti-commutations relations on these creation and annihilation  operators.  For fermions these read as here where the curly bracket denotes the anti-commutator.  And these relations induce via these isomorphism symmetric bilinear form on the number space.  So in total I got a symmetric bilinear form and the real structure and of course I can  then can clearly define in a product by combining these two informations and the take a message  from this slide is basically that the number space is a real Hilbert space just by construction.  So here's also essential that we are considering fermions because for bosons we would have  to replace the anti-commutator by the commutator but this would lead to minus sign here and  here and then we don't have a positive definite inner product so that's no longer Hilbert  space.  So yes?  What is the isomorphism?  Is it just the obvious?  Yes it's from here you can see that this is linear.  The argument so you can map this to V but this is anti-linear in the argument because  here you have the first isomorphism so I have to identify with the dual in order to get  a linear isomorphism.  So usually this one is not canonical.  What is not canonical?  The resismophism is canonical if you have an inner product on a Hilbert space.  Yes I need the inner product on the single particle Hilbert space.  Yes?  Yes there's.  Yes it's an even.  Why would you get one to odd dimension?  No because you said the non-phosphate is a real dual space so you really must have some  even dimension, generalized even dimension within the structure.  Yes I mean.  So you can't get a situation.  So the real dimension is even yes.  With the complex dimension.  Okay so now operators on this number space so general operators in the magnetic particle  picture with interactions would just be contained in the Clifford algebra over W as Chris for  example considered.  So the CRI algebra of course you have to need to have further conditions like even parity  and stuff like this and the Q is just the quadratic form that is induced by the car form.  But I want I only can consider non-detecting operators and these are generated by these  quadratic order operators and one can easily show that these are isomorphic again by linear  isomorphism to the linear operators over my number space.  And if I only want to consider single particle operators with fixed particle number then  I would only allow these two and these would correspond to really indeed linear operators  over my single particle space.  So then I would recover the usual picture of single particle physics.  Okay so the crucial observation is now that time evolution has to be unitary and orthogonal.  I think unitary is well established assumption and orthogonal means here that the unitary  time evolution has to preserve the anti-commutation relations between the creation and annihilation  operators.  So that's also a reasonable assumption and together this means that all Hamiltonians that  are physically reasonable have to be self-enjoyed and skewsy metric or post differently they  have to be self-enjoyed and imaginary with the real structure that is induced on the  operators over the number space.  So that's now fixed and this is my starting point for my classification.  Now in the classification in the bike as I said before I make the fundamental assumption  that the bike is kept.  So I can consider always the flattened Hamiltonian that squares to one.  I have one question there.  Yes?  Non-interacting setting needs to be enforced already at that level or?  In order to consider operators over the number space it's necessary.  Unitary evolution needs to be orthogonal.  Well the complex over the space would be double-dimensional.  Yes.  So I was wondering how you were allowed to speak for generic interaction.  No, no interaction.  This Hamiltonian is in the sense of Hamiltonian before second quantization.  Which one?  Right.  The one on the previous slide.  It's a general Hamiltonian over the number space.  Yes.  That's the one.  That's the one.  Still state space is some sort of second quantization of number space.  Right.  But in real representation that's why you...  So this observation you cannot move it.  After the first thought you need to have a mainstream...  Yeah.  I didn't thought about this in detail yet.  Okay.  So the aims of the next section are now to establish first of all systematic description  of this order of solid state systems in this number space picture.  And then derive a systematic and constructive.  That was one of our central motivations to make everything constructive so that in the  end you indeed can calculate stuff and do this for a symmetric crisis of the tenfold  way.  Okay.  So in the bike the single particle space as said before is A2 of ZD.  So any bravado letters with some internal space.  And the corresponding number space is B plus B dual.  So I can write it in this form.  And similar to myos talk I introduce finite range operators as written here.  So it's just finite hopping of operators.  And the C star that is generated by all these finite range operators is the mathematically  equal to the uniform row C star as observed by Kubota.  And I denote this row C star algebra by this type here.  Okay.  So in principle one can start with this C star algebra but as we learned yesterday the  K theory is not very treatable.  So we have to go further.  And instead of going to the non-uniform row C star algebra it was one possible way I introduce  this order in my systems by using the techniques by Belissar and many authors that are here  in the audience.  And this goes as follows.  I consider a disorder configuration space.  For simplicity I assume that it's of this form.  So omega naught denotes some kind of disorder on each unit cell.  And I copy it for each unit cell.  So I got a canonical z, d action by translations on this disorder space and for technical reasons  I also need a probability measure.  But with this I now consider the so-called disorder observables which are defined as  all the functions from my disorder space into the set of allowed operators over my letters  such that this covariance condition is satisfied.  This covariance condition can be thought of as a tautological condition because the disorder  breaks the translation symmetry.  So if I consider one fixed disorder configuration and shift it this should be the same as shifting  the underlying letters.  So this has to be satisfied and thus we make this assumption here.  And this will be the set of all disorders observables over the bike letters.  Now the use of the translations over the letters.  So lambda is the, yeah just, yes exactly.  Yeah so I translated everything into linear operators over my number space so that I don't  have to consider c and c dagger anymore because mathematically it's the same.  So that's why I don't use this here.  And the name homogeneous disorder can be understood by this observation here that for any given  t in my disorder observable algebra I can average over all disorder configurations and  this yields a translation invariant operator.  So if I go zoom out very largely out of my solid then I would recover a translation invariant  system again.  So and no disorder is simulated by setting the local disorder to a point.  And then this is just translation invariance.  Okay so there's a theory in Barrow from 2003 that if I am in the clean case then this c-style  algebra is just the usual c-style algebra of translation invariant operators.  So I can consider the torus again and do the usual classification.  Now in our setting we have a general, more general theory that if I consider any disorder  configuration space then I can show that our c-style algebra is isomorphic to the usual  c-style algebra, the cross product c-style algebra that is used by Belisar and Fodontovs  Byser and many other authors.  And this cross product c-style algebra can be considered as the closure of all polynomials  of this form.  So the u again denote the shifts along the letters and the m is now a disorder valued  local parameter.  So by these natural physical constructions I in particular showed that I always can write  my hematomines in such a polynomial form.  Okay.  All right.  Now if there are no questions to this I will now go on to explain symmetries to you.  Of course I want to consider all symmetry classes of the tenfold ways so I have to explain  to you what I mean by symmetries.  The symmetry we define it as a real so it preserves the structure of my real hibbert  space unitary or anti-unitary.  This means anti-linear isomorphism that commutes with the Hermitonian.  This differs slightly from other definitions of symmetries but we only allow symmetries  to commute with Hamiltonians.  And in this talk it is sufficient to consider only local symmetries that only act on the  internal space so this w.  My physical examples of local symmetries are of course time reversal symmetry.  Initially it acts only on the single particle hibbert space but by my construction to the  number space it lifts to an anti-linear operator over my number space which is of this form.  Then I also consider charge number conservation.  This corresponds to the symmetry iq where q measures the number the charge is the charge  operator.  So q is 1 on particles and minus 1 on holes because they have opposite charge.  And I have to insert the i here because I demand that all operators are real or symmetries  but of course my real symmetry comes from part of your whole conjugation and charge  anti-commutes with part of your whole conjugation because holes have opposite charge than particles.  And another one is particle hole transformation so this is an anti-linear operator that almost  looks like my particle hibbert conjugation but it is enriched by some unitary which acts  on a single particle hibbert space and squares to one.  Okay, these are some examples and now in order to perform a classification I could in general  try to classify all flattened Hamiltonians in age but from a mathematical point of view  these is a bit difficult because the Hamiltonians are imaginary so it's quite unnatural to consider  them.  Instead it's much more practical to consider the corresponding real operators which we  call quasi-particle vacua which is just an i in front of the age in the minus sign of  convention.  So then of course I have to translate Hamiltonians with these local symmetries into quasi-particle  vacua with symmetries and these symmetries are called pseudosymmetries which is introduced  by Kennedy and Zambor.  And these are defined as follows, a quasi-particle vacuum here the Riemann under what it contains  as algebraic conditions has r positive and s negative pseudosymmetries if there's a  clifford action into my algebra such that I have r operators that are self-adjoint and  square to 1 and s operators that are skew adjoint and square to minus 1 and all of these anti-commute  with my j.  So the small k and small j denote the real generators of my clifford algebra.  Okay, and my claim is that all symmetry classes of the tenfold way can be written, can be  represented in this form.  So just an QPV that represents the ground state plus a clifford action.  And that's what I want to explain now.  So let's consider only the real symmetry classes, the complex symmetry classes are already  explained in very detail in the book by Proudhan and Schultz-Biles for example so I won't consider  them here.  So let's start with class D, class D corresponds physically to a superconductor with no symmetries  so I also got no pseudosymmetries and that's it.  So class D3 corresponds to a superconductor with time reversal symmetry and time reversal  symmetry as explained is an antilinear isomorphism on W which commutes with my h but is antilinear  therefore it anti-commits with j which is just i times h.  So I can construct exactly one pseudosymmetry by this here.  Okay, so the next class is A2, A2 is a topology insulator because I have charged conservation  with time reversal symmetry.  And there I can again define this one but furthermore I can define another one by defining  J2 equal i gamma, gamma was the real structure on the number space times t times q and one  can verify that this indeed anti-commits with j and squares to minus one and the skew here  mission.  Okay so the next is a topology insulator with time reversal symmetry and a particular  Holtz transformation one may also call it a chiral symmetry and then one can do similar  steps and define a third pseudosymmetry and so on and so forth.  Maybe class, I also mentioned class C because it's not the usual form on encounters in the  literature for the symmetry classes.  So class C is a superconductor, so no charge conservation with spin rotation symmetry.  Spin rotations are generated by three operators corresponding to the polymetrices but again  I have to put an i here in order to make them real and they commute with h if my hematoma  is spin symmetric.  And then I have to go to two by two matrices in order to define four pseudosymmetries and  these are then of this form and my j has this form and yeah.  H and minus h yeah.  Okay so that's a little bit different but the other cases are then formed in a similar  way just by defining the pseudosymmetries that I explained before in this block diagonal  form.  So and these are all eight classes then translated to QPV with species symmetries.  Okay so now after all my physical input is some QPV and some input of symmetries that's  all and I have a of course a fixed index of positive and negative species symmetries and  now I want to consider the set of all allowed QPV such that this is a symmetric QPV.  And I start my classification by considering all homotopy equivalence classes in this set  of allowed ground states or QPVs.  If I only consider homotopy theory I could stop here and just calculate this set but  I want to do k theory and therefore I want to stabilize and in the first step I then  explain what I mean by QPVs over matrix algebras over a so if I get more bands in my system  then I define it in this way where phi n denotes the direct sum so the diagonal action of phi  and then of course I can consider homotopy equivalence classes on this larger algebra.  Now in order to compare QPVs in different matrix algebras I have to somehow explain how  I do this and this is done by fixing some ground state or QPV which I call topological  trivial.  I have to fix this and then I can say okay my QPV in my smaller algebra is the same as  the QPV in a larger algebra if it's just a direct sum with some topological trivial  summand.  And this forms an inductive system and then if I go to n to infinity so I allow any degrees  of freedom added then I finally obtain a set of stable homotopy equivalence classes and  then my set of topological phases is defined as the set of all topological phases I defined  in this way but always relative to each other.  This means I only can tell you what are topological phases relative to each other.  I have no absolute notion of topological phase.  This is only fixed by this J-Ref.  Okay and then okay maybe I just this is a bit technical I will skip this.  The only take home message of this is this physical motivated set of topological phases  can be identified with van Dalles K theory where van Dalles K theory is built upon odd  self adjoint unitary matrices but since I have not much time I will skip this here and  just tell you that I can identify my group of topological phases with K theory via this  identification.  So in the end I obtained a canonical definition of topological phases for some QPV relative  to some reference QPV in this way.  And yeah okay so after all I have a new motivation for the usual cross product C star algebra  as the algebra of observables and yeah my definition of a topological phase only depends  on R and S so maybe on my phi here so it doesn't distinguish between the different symmetry  classes.  And this is a bit hard to understand at the beginning that I have to fix a trivial topological  phase but it's like this and I can identify with K R theory.  Alright and as an advertisement with the systematic constructions I also can now define systematic  sets of indices.  So I obtained that my topological phases live in such a K theory group here and for any  given index R, S I can define an index so this is only a Z-valued index by this formula  so it's an adapted formula for the complex cases it's a so-called conjourn sorry character.  So this is just an advertisement it's not possible to explain it in a way that it's  completely understandable but so just yeah.  Not yet.  Yes.  You're fine in K R, it's just how you get a number.  Is this pairing only these non-torsion invariants?  Yeah currently I'm trying to use Kehl-Endon's results in order to also derive formulas for  the torsion values but that's a bit more technical and more difficult.  Yeah.  This these J's?  Yes.  So you always insist that they must be from the algebra.  Yes.  But they could be the automorphism group.  And it's not possible in general to directly identify them with K R theory I think.  So yeah my assumptions are essentially based on this because yeah.  I mean you're constructing the J-1 to J-S from the symmetry operators so I mean they're  just in the algebra.  I think they're even point wise right so they were just a matrix algebra and then everything  is in a any way.  Yeah right in this case yes.  Not local symmetry where restructure is mixing things at different places.  If you really want it.  Well I could extend my notions to non-point wise things but in the end I have here a projection  so this is a corner algebra and what I need to in order to identify with the K theory  of this inner algebra I need to have that the K theories are the same.  So for this I'm not sure how this works for outer symmetries.  But I think that you need to extend this one on the fox base.  Okay so now shortly the top of the classification at the boundary.  So I introduce a boundary as done often in the literature by just considering an infinite  boundary perpendicular to one translation and direction.  So in this way I obtain the half space letters which is just the right hand letters here  and the boundary letters is still ZD so it's all respects the translation and directions  in the ZD minus one direction and the primary difference to the bike is that my shift operators  perpendicular to the boundary are no longer unitaries because of course if I shift something  over this boundary I map it to zero.  So I obtain this condition here where P0 is the projection on this edge here this boundary.  But besides this fact I can now define the algebra absorber with over this half space  letters in the same way as in the bike again I have this covariance condition and here  I put in the half space letters and the same for the boundary.  Yeah there again I don't have a head because all shifts in this direction are still unitary.  But basically the definitions are the same and I can show that the half space algebra  is now isomorphic to the cross-product C star algebra with the semi-group ZD minus one  times N and this can be considered as the closure of all non-commutative polynomials  of this form where this pre-factor is now in the boundary algebra.  So all shifts are contained in here with a parallel to the boundary.  Alright so now what's with the QPVs?  Yeah first of all I want to explain to you the canonical morphism from the half space  to the bike this is important to distinguish in the end between boundary observables and  bike observables in the half space.  So and this is done by a bike projection from a half space which just maps the half space  translation operators onto the bike translation operators and in this way I obtain a set of  boundary observables that are localized at the boundary and these are defined as all  operators that are in the kernel of my bike projection.  So all observables that are vanish if I go into the bike have to be at the boundary.  So and this is the idea that is generated by the boundary projection and this can be  shown to be isomorphic to this C star algebra.  Okay and as I said here this is the observation physically and I can put this information  in a shorted sequence so these are the boundaries this can be identified with some set in the  half space and on the right hand side I got the bike and these all together from the half  space.  I don't think you define what this A delta actually is.  A hit?  No A delta.  I mean I know that you say that it's isomorphic to this kind of thing.  This one here.  Okay so this is the D-dimensional version of the bike I just showed you.  Bike algebra.  No.  Yes here it's important that the symmetries are on site.  Yes.  Yes indeed.  So the symmetries are on site but the J is not on site so the J cannot be unitary in  general anymore.  So of course I have gap less boundary states.  There's no gap anymore.  So what I only can demand is that when I project into the bike then I have a gap operator.  That's my definition of a QPV with boundary.  So that when I project in the bike I recover a QPV.  And then I can follow by general properties of my boundary projection row that J has to  be a QPV so it has to be squared to minus one up to boundary states.  And this is the operator encoding of gap less boundary states.  And the unitary star morphism since it's act locally is still a unitary star morphism  now in the half space algebra.  But I cannot demand anymore that it's exactly anti commutes with J because yeah this does  not have to be satisfied in general from an algebraic point of view.  And there's a canonical association between bike QPVs and half space QPVs by a section  of this short exact sequence.  Now I would like to establish some kind of topological classification.  So some kind of homotopy equivalence between QPVs with boundary.  But I have in my mind that I want to classify the properties at the boundary.  So if I just consider a homotopy of operators in a head this would still contain some informations  about the bike because the half space contains both the boundary and the bike.  So I have to somehow lose my conditions on my homotopy.  And this is possible by going over to the so called multiplier algebra over my localized  boundary states.  And this can be defined as all operators over the half space.  Here I still have the disorder in this picture that preserve the boundary states.  So this means that this T is times some operator in this boundary states.  This is still a boundary states.  In particular these are still covariant with respect to the translations parallel to the  boundary.  In other words this means that I preserve the covariant conditions parallel to the boundary.  And I no longer demand that my operators in this algebra are covariant with respect to  the perpendicular translations.  So in my opinion this is a reasonable assumption because a boundary destroys translation invariant  by construction.  So it doesn't make any sense to assume this covariance if I'm interested in properties  at the boundary.  So that's why we dropped this condition.  And furthermore there's an integrity, more stomaphism from my general algebra into the  larger algebra.  And yeah, so this now generalizes the QPV with boundary to any set of two builds where  F is now satisfies the same conditions as J.  But the only difference is that it's now an element of the multiplier algebra.  And my phi now also maps into the multiplier algebra but I still have the same conditions  as before.  And I will call such a topological trivial, oh there's a typy here, there should be F's  here.  Sorry for that.  I call such a trivial if all of these conditions are equal to 0 of these quantities.  So if I still got exact symmetries my J hat is still skew Hermitian but crucially my J  hat squares to minus 1.  And as we learned before, if J hat squares to minus 1 this means that there are no gap  plus boundary states.  So from a physical point of view this is a natural notion of topological trivial phase.  And under those by definition this set defines a K group by modding out these topological  trivial phases and considering only homotopy equivalence classes of these two builds.  So in this way I have a second picture of topological phases which is again a K group  but now from a completely different point of view.  And to summarize now the boundary topological phase of any disorder topological insulators  or superconductor is defined as the equivalence class that is generated in this set of topological  phases at the boundary.  And some algebraic properties that follow by these constructions are for example that  I always can demand that my symmetries are again preserved exactly so I can demand that  my symmetries are still anti commuting with J hat and I also can demand that my J hat  is skew Hermitian.  And so the only fact that prevents topological phase from being trivial is that it's not  in general squares to minus one and if it does then it's trivial.  And as I said before the section of my boundary Schottx sequence associates to any given bike  QPV a canonical boundary QPV with boundary.  And to summarize from a physical point of view two boundary topological phases are equal  if and only if there's a path in this multiplier algebra such that these are two are equal up  to trivial phases.  And furthermore a boundary topological phase is trivial if and only if there are no stable  stable in the sense of homotopy classes, Gapless boundary states.  So yes that's another observation here the reference is already there it's the one where  there are no Gapless boundary states.  Why?  I don't quite get the picture.  Yes.  But these are two are private two independent half spaces.  So.  So there is the folding trick where you fold it and then you reuse interface the backroom.  Okay.  Okay.  How much time?  Yeah, I can either make a short or I mean I already have explained some to some extent  the topological phases and now the bike boundary constant is quite easy to construct because  yeah from a mathematical point of view the bike boundary constant is established on the  K theory.  I have already introduced the bike boundary short exact sequence here as a reminder this  is the set of observables that are localized at the boundary half space and the bike and  this induces always a long exact sequence on the K theory which is of this form but  the only relevant morphism for us is the one that is here diagonal.  It's the one that maps bike topological phases onto boundary topological phases and I can  interpret it like this because I have explicitly identified my topological phases with K R  theory groups and thus I immediately get a bike boundary cosponence by this morphism  which is constructed from abstract mathematical concepts.  And now the main result of our work is that this bike boundary morphism maps topological  phase in the bike which is measured with respect to some J ref onto the boundary topological  phase and again here occurs the reference KPV.  This might surprise you a bit but if this J ref is for example an atomic limit topological  insulator or superconductor then this phase is trivial so then indeed I only get this  class here so what one actually should expect but of course if I consider here any possible  difference of two KPVs then the difference has to be observed at the boundary too.  And these J heads are the canonical lifts from my bike to my half space.  Okay so some, finally some physical properties.  The image of the bike boundary cross bondence is equal to the set of all boundary topological  phases if my omega naught is contractable but this doesn't have to be true in non-node  more general settings of this order.  Or stated differently all boundary topological phases are corresponding to some bike topological  phases so I cannot just change the boundary without changing the bike.  So one physical formulation of bike boundary cross bondence.  And another point is given a fixed bike class the boundary class depends on the direction  of the boundary.  Probably most of you will notice but for me it was a bit surprising that you measure different  topological phases depending on which phase you cut your system in.  It's not all the same.  It's only the same for the strong topological invariant.  This is the same on all boundaries up to a sign which depends on some orientation of  your space but this is seen on all phases and this also applies that if I got a non-trivial  strong topological invariant in the bike then all boundaries have gapless localized states  which cannot be destroyed by local.  Well the bike is gapped so I cannot move any gapless state with small amount of energy  into the bike.  So the mathematical statement is I have always states in the gap.  Right?  But they don't specularly localize that.  Localized.  So that's what you mean right?  They're very much stretched along the bike.  They're very non-local.  Localization is still there.  Yeah okay.  They're transversely localized.  Yeah the mathematical assumption was in the bike I have such a gap spectrum and the statement  is if I introduce a boundary then I get some gapless spectrum and since I know it cannot  be in the bike it has to be at a boundary.  That's how I think about it.  Yeah.  Yeah?  To make a statement that this is what we call the spectral statement and there is a dynamical  statement on top of this which is that these gapless boundary states do not understand  localize.  What we found is for sure you need pairing to go all the way into index theory to resolve  the second layer of the problem.  Okay.  Which is the dynamical statement.  Okay.  So that's why the word for your harness is pairing is quite important.  Okay yeah.  But yeah.  Are you finished with the work yet?  Yeah.  So let's give it a hand.  Yeah.  Thank you.
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