Kitaev's Periodic Table (of Topological Insulators & SCs)

[Kirjava]

Hi PF,

I'm trying to come to grips with the work of Alexei Kitaev on applying notions from (topological) K-theory to the task of classifying phases of topological insulators and superconductors (paper here: http://arxiv.org/pdf/0901.2686v2.pdf). Despite having plenty of citations, I've yet to find a single review which talks about his construction in any sort of detail (K-theory isn't exactly part of the physicist's working toolkit in condensed matter so I'm not terribly surprised). It's a bit of a long shot, but I wondered if there were any physicists (or mathematicians) around here familiar enough with this work and willing to answer a few questions?

 

[Physics Monkey]

I may be able to help a bit, if you're still interested. You may also find some information in here http://arxiv.org/abs/1005.0583

 

[Kirjava]

Hi Monkey, thanks for the reply and the paper. I'm still just getting acquainted with the field of topological insulators and condensed matter in general, so my questions are likely to come across as quite amateur (they are). One that's been bugging me is the following:

In both Kitaev's paper and the one you linked the authors restrict attention to fermionic Fock space Hamiltonians which can be written in terms of Majorana operators. This may be necessary to make progress via the classifying space approach in your paper, but I don't see why it's necessary from the perspective of K-theory. Can we be more general by allowing an arbitrary (gapped) Hamiltonian in the Fock space (possibly containing quadratic terms and higher), or is this physically nonsensical for some reason that I'm missing?

 

[Physics Monkey]

If I understand your questions correctly, you've touched on two separate issues.

The first issue is the role of majorana fermions. I think the crucial point is that any fermionic fock space operator can be written in terms of majorana operators. Physically, majoranas are useful because they allow one to treat conventional hopping and superconductivity on the same footing and lead to a simple unified form for the hopping hamiltonian. This is the same in all approaches, independent of K theory considerations.

The second issue is the role of interactions or equivalently, the assumption of quadratic hamiltonians. Certainly in principle we would like to treat arbitrary gapped fermion hamiltonians, but this is a much more challenging problem (although we have partial information and kitaev himself has new unpublished results in the area). The classification in kitaev's current paper and the other topo. ins. classifications are all for strictly non-interacting phases. Some of these phases survive the addition of interactions, but others become equivalent once interactions are included e.g. http://arxiv.org/abs/0904.2197

Does this address your question?

 

[Kirjava]

Ah, I meant to say " ... written in terms of quadratic combinations of Majorana operators", so your second point was really the one I was after. Sorry about that.

It seems to me that if we relax the restriction to quadratic hamiltonians the "real structure" of his K-theory classification is lost (the idea being that whereas before we were restricted to real anti-symmetric matrices after switching to Majorana operators, the most general Hamiltonian operator will be given by some large hermitian but otherwise arbitrary matrix acting on the full Fock space). So perhaps in this general setting there is only one large complex class? (with a much less interesting structure of invariants given the 2-periodicity of complex K-theory). I'm sure Kitaev is cooking up something much better than this, but would you agree with this reasoning?

 

[Kirjava]

Another less speculative question if anyone is still out there:

I've seen a few authors talk about creation/annihilation operators as though they're automatically defined globally over the whole Brillouin zone (e.g. Equation 12 of the paper Monkey linked). Doesn't this amount to assuming that the BZ vector bundle of states is a trivial bundle? Do we not therefore lose some relevant topology by proceeding in this way?

 

[Physics Monkey]

Ah, I meant to say " ... written in terms of quadratic combinations of Majorana operators", so your second point was really the one I was after. Sorry about that.

It seems to me that if we relax the restriction to quadratic hamiltonians the "real structure" of his K-theory classification is lost (the idea being that whereas before we were restricted to real anti-symmetric matrices after switching to Majorana operators, the most general Hamiltonian operator will be given by some large hermitian but otherwise arbitrary matrix acting on the full Fock space). So perhaps in this general setting there is only one large complex class? (with a much less interesting structure of invariants given the 2-periodicity of complex K-theory). I'm sure Kitaev is cooking up something much better than this, but would you agree with this reasoning?

This can be true in some cases, but there is an important further constraint on the many-body Hamiltonian provided by locality.

There are at least two relevant notions of locality.

1. Geometric locality: we require that the Hamiltonian only contain interactions between nearby particles or spins.

2. Bounded interactions: we require that the Hamiltonian only contain interactions that couple a bounded number of particles together e.g. 5-body interactions are ok but N-body interactions are not (N is the number of particles).

Even if you relax 1, for example by studying a system on a complete graph, 2 is still a strong contraint on the types of Hamiltonians you can write. The resulting problem could then be to study the space of gapped local many-body Hamiltonians up to stable equivalence.

 

[Physics Monkey]

Another less speculative question if anyone is still out there:

I've seen a few authors talk about creation/annihilation operators as though they're automatically defined globally over the whole Brillouin zone (e.g. Equation 12 of the paper Monkey linked). Doesn't this amount to assuming that the BZ vector bundle of states is a trivial bundle? Do we not therefore lose some relevant topology by proceeding in this way?

Yes, this is mostly just laziness I think. The simplest example is the Chern insulator in two dimensions. The momentum space connection has a non-trivial first Chern number for the lowest band, so the bundle cannot be trivial. This is related to the better known statement in real space that there are no localized Wannier states using only orbitals from a band with non-zero Chern number.

On the other hand, if the microscopic model is a lattice model with a finite number of states per site, then the full set of bands is always trivial in the appropriate sense.

 

[Kirjava]

Thanks again Monkey, some resolution on these issues comes as quite a relief (unfortunately nobody in my department studies these things, so they'd been bottled and stewing for quite a while). Rather than harassing you with endless questions though, could I ask you to recommend a text which might help to fill some of the gaps in my background? I'm a final year undergrad studying Kitaev's paper (the one I linked above) as part of a senior thesis, and my exposure to CMP has been relatively limited so far. I know of classics like Kittel, Ashcroft and Mermin etc. but these are relatively old and haven't been terribly helpful yet. Kitaev makes the mathematical pre-requisites for his construction quite clear, so that hasn't been a problem. He is obviously writing for a seasoned CM audience though, and that's been giving me a bit of trouble.

Edit: to clarify, I'll be interested at some point in reviewing our understanding (or lack of) how Kitaev's picture breaks down in the presence of interactions. I get the impression that Monkey's two points above regarding this issue are sort of 'common knowledge' things (no? I suppose the first is quite intuitive but I don't know about the second). Is there an obvious place to go in search of more 'common knowledge'?

 

[DrDu]

I don't really understand this stuff but maybe some review article may be helpful to understand the physical background:
http://arxiv.org/abs/1002.3895

 

[Physics Monkey]

No problem. I really don't mind discussing with you, it's a nice change of pace from questions about time dilation, etc. and not so many people are interested in this subject anyway.

I will think a bit about the best resources for background, although unfortunately I lot of this stuff is pretty new and not well documented in a pedagogical way. To help me make suggestions, may I ask if you prefer more mathematical versus more physical arguments? (Many of the players in this area are part mathematician.) Also, do you have any interest or background in quantum information and quantum computation? (Ideas from these areas play a non-trivial role in the background to this stuff.)

 

[Kirjava]

That time dilation, how does it work?

In seriousness though,

To help me make suggestions, may I ask if you prefer more mathematical versus more physical arguments? (Many of the players in this area are part mathematician.)

Personally I've found many of the more 'physical arguments' in this area to be quite baffling. Again, they presume a certain amount of familiarity and intuition for concepts that I just don't have. The mathematical arguments can be equally baffling, but the necessary concepts are often in plain sight. So the mathematicians have that in their favour. I also want to get on to reading Shinsei Ryu's paper connecting this area to D-branes (http://arxiv.org/abs/1001.0763), and K-theory seems to be a common element in that relationship. This isn't how I'd approach the subject if I were tackling it on my own, but I have a thesis to write and my advisor's expertise is in strings rather than CMP.

So regarding your second question: I have no background, I do have interest, but then I don't have quite enough time. Then again, if some appreciation for the issues relevant to my line of investigation is within reach I'll certainly go for it.

 

[element4]

I'm also very much interested in discussing these papers, although sadly I am quite busy currently and might not be able to participate too much.

Let me add some references that might be useful. Stone et al (J. Phys. A: Math. Theor. 44 045001) has clarified certain aspects of the Kitaev paper and is a good place to start (see also this). There are generally many sources where the usual K-theories are discussed, but its harder to find good references for KR-theory (which is the relevant thing, when assuming translational symmetry since the symmetries act as involutions on the bundle over momentum-space).

K-theory comes into play in string theory in the problem of classification of stable D-branes and Ramond-Ramond charges (it seems that there are stable non-BPS D-branes, these are captured by the K-theory classification but missed by the SUSY analysis). It is from here Ryu and Takayanagi tries to connect D-branes to topological insulators/SC's. A review of the string theory side of the story, and basics of K-theory, is given by Olsen and Szabo (arXiv:hep-th/9907140).

 

[Kirjava]

Hi element4, thanks for the links. Regarding sources for KR-theory, I've found that a combination of Kitaev's primary reference (the book by Max Karoubi) and the original paper by Atiyah (e.g. www.maths.ed.ac.uk/~aar/papers/atiyahkr.pdf) is fairly comprehensive. Karoubi develops ordinary K-theory in a rather general (categorical) setting so that it's fairly clear how the theory extends to the category of real vector bundles (he relegates the details to exercises in fact - these are the only places in the book where KR theory is mentioned).

So perhaps I'll try to summarize what I understand of Kitaev's scheme so far, and anybody who's interested can chime in with comments, corrections or questions:

The situation is basically that we're given a fermionic system (e.g. a lattice model) and we want to classify its gapped non-interacting phases. In more mathematical words, we want a classification of gapped Hamiltonian operators acting in the Fock space of single-particle states which satisfy the constraint of being no more than quadratic in the fermionic creation/annihilation operators. If the system has some type of translational symmetry, then it makes sense to talk about Hamiltonians defined on some momentum space (be it a Brillouin zone torus or whatever else). We will further assume that the Hamiltonian is irreducible with respect to non-trivial unitary symmetries (for a system that isn't we'll then obtain a classification for each of its irreducible components).

Now at each point on the momentum space we have a vector space of one-particle states. This defines a family of vector spaces, and I've managed to convince myself that this family should be at least locally trivial (although it's crucial in some cases that global triviality is not guaranteed, as Monkey pointed out earlier). Our Hamiltonian is not exactly a one-particle operator however (even though we're not including serious interactions) since it can contain terms of the form [tex] a_i a_i, a^{\dag}_i a^{\dag}_i [\tex] (corresponding to pair-creation/annihilation presumably?). So it doesn't properly operate on the one-particle bundle. Instead its natural place is as an operator on the 'Nambu space' or 'particle-hole' space defined for e.g. on pg 4 here: http://arxiv.org/abs/1101.1054 (Kitaev doesn't talk about this explicitly but I think it's equivalent to what he's doing with majorana operators). In our case we have a 'Nambu bundle'.

Since we want our classification to be stable to continuous deformations, we're only interested talking about Hamiltonians acting on this bundle up to homotopy. We can therefore 'flatten' H's spectrum by taking all positive eigenvalues to 1, and negatives to -1 (so that H^2 = 1). How we proceed from here depends on the presence or absence of any 'unusual' symmetries. The irreducibility of H wrt ordinary unitary symmetries implies that we need only consider at most one 'Time-Reversal' type (anti-commuting) anti-unitary symmetry, and one 'Particle-Hole' type (commuting) anti-unitary symmetry. If there are no such symmetries then all we have is a complex vector bundle over our momentum space, and we seek to classify automorphisms satisfying H^2 = 1 up to homotopy. There is one final element here in that K-theory only deals with these questions up to 'stable equivalence', which means that when comparing two systems we're allowed to augment each with an arbitrary trivial system. Whether this should be viewed as a feature of the classification or a defect is unclear to me (I can see that adding a trivial system might have some physical justification in terms of adding a few states with trivial hopping elements, but others have referred to stable-equivalence as a limitation of the K-theory approach). With this caveat though our question is answered precisely by the 'zeroth' K-group of the momentum space according to Karoubi's presentation of the groups K^p,q (I won't go into detail about this right away but I'll do my best to answer questions if anybody's interested).

Extending this classification to cases where there are additional symmetries to respect (possibly involving involutions on the momentum space e.g. time reversal) is a matter of employing the additional machinery of KR theory. I think I have a handle of most the details involved, but I'm more interested in getting the physical picture right for now.

There is more to Kitaev's paper than all this. He gives a classification of discrete systems as well, and talks quite a bit about Dirac operators. I haven't made much progress on understanding these aspects yet, but if my picture of the rest is reasonably complete then they're next.

 

[Physics Monkey]

That time dilation, how does it work?

In seriousness though,Personally I've found many of the more 'physical arguments' in this area to be quite baffling. Again, they presume a certain amount of familiarity and intuition for concepts that I just don't have. The mathematical arguments can be equally baffling, but the necessary concepts are often in plain sight. So the mathematicians have that in their favour. I also want to get on to reading Shinsei Ryu's paper connecting this area to D-branes (http://arxiv.org/abs/1001.0763), and K-theory seems to be a common element in that relationship. This isn't how I'd approach the subject if I were tackling it on my own, but I have a thesis to write and my advisor's expertise is in strings rather than CMP.

So regarding your second question: I have no background, I do have interest, but then I don't have quite enough time. Then again, if some appreciation for the issues relevant to my line of investigation is within reach I'll certainly go for it.

Well, Kitaev's masterpiece http://arxiv.org/abs/cond-mat/0506438 has a lot of good background information in it although there are only glimmers of the K-theory approach here (if you know where to look). It is more about what we would call long range entanglement and topological quantum computation, but some of the basic ideas are still very useful. This talk http://online.itp.ucsb.edu/online/topomat11/kitaev/ is also useful and closer to the K-theory stuff. Importantly, you can hear him talk about the crucial distinction between long and short range entanglement (K-theory is for the latter). There are other approaches that give the same answer as Kitaev's K theory classification e.g. random matrix theory, group cohomology, but it sounds like you don't have time for these things.

As an aside, one other place where K theory has appeared is in a sort of classification of Fermi surfaces due to horava.

The real reason why K-theory appears (in both the D-brane and cond-mat contexts) is our interest in stable equivalence. Basically, we want to be able to add "trivial" degrees of freedom without changing anything essential (so long as the energy gap is preserved). In string theory this means brane-antibrane pairs which carry vector bundles on them. In the context of free fermion models this means adding arbitrary gapped flat bands carrying trivial bundles of bloch states over momentum space.

Another aside, nothing against Shinsei's paper, but one should be aware that the physics is quite different for the cond mat systems we normally think about and the d-brane systems he considers. Roughly speaking, the cond mat system is like a small subset of the degrees of freedom but a lot more is going on. At some level the connection between these two papers is just that 8 = 8.

 

[Kirjava]

Another aside, nothing against Shinsei's paper, but one should be aware that the physics is quite different for the cond mat systems we normally think about and the d-brane systems he considers. Roughly speaking, the cond mat system is like a small subset of the degrees of freedom but a lot more is going on. At some level the connection between these two papers is just that 8 = 8.

I've been wondering about where this connection would lie along the axis of profundity, but figured that I wouldn't find out until I got around to studying it in detail. Hmm. Thanks for the references again. I've also been entertaining the possibility of investigating the status of Kitaev's work within the more natural context of topological orders, rather than jumping to strings. Maybe a bit of both (and less sleep) would be the best compromise.

 

[ComradeVVA]

The irreducibility of H wrt ordinary unitary symmetries implies that we need only consider at most one 'Time-Reversal' type (anti-commuting) anti-unitary symmetry, and one 'Particle-Hole' type (commuting) anti-unitary symmetry.

Hi everybody,

It's great this thread is here and hope you guys can see this beacon for help. I'm confused about what the above quote refers to in Kitaev's paper.

As far as I understand, Kitaev elaborates on four cases for a generic free-fermion system for d=0, which in first-quantized form is just a skew-symmetric matrix A:
1. No symmetries: classifying space R_2
2. A unitary commuting U(1) symmetry, represented by Q with QA = AQ: C_0
3. An antiunitary anticommuting discrete symmetry, represented by T with TA = -AT: R_3
4. A combination of cases 2 and 3, which implies another antiunitary anticommuting discrete symmetry: (QT)A = -A(QT): R_4

So I don't understand the above quote (and all those other papers that construct these classes in terms of particle-hole symmetries). Which operator (Q,T,QT) is the antiunitary commuting particle-hole symmetry? Similarly, why is Kitaev's R_2 case (class D) described as having no symmetries while Furusaki's (10.1103/PhysRevB.88.125129) and Ludwig's (doi:10.1088/1367-2630/12/6/065010) papers state that that class has a particle-hole symmetry?

I greatly appreciate any and all help; may acknowledge discussions if I ever write up related stuff.

- VVA

 

[Kirjava]

Hi Comrade,

I'm glad to see that this (rather old) discussion is still proving useful to somebody. I started this thread whilst I was (beginning) to write my undergraduate thesis on the subject of topological insulators and their K-theory classification - out of curiosity may I presume to ask what your interest is in this area?

As it turns out I eventually found it helpful to put Kitaev's derivation of the periodic table aside and work out my own using the basic machinery of K and KR (and KH) theory. The result was (I think) a more systematic derivation of the periodic table which cleared up a lot of my confusions about Kitaev's approach.

To answer your (boldface) question however: by using a basis of Majorana operators Kitaev has made the "particle-hole" symmetry (the one referred to in your other papers) particularly trivial: it's nothing other than complex conjugation itself. So the particle-hole symmetry is there in the form of a reality constraint. Another approach would be to represent the free-fermion Hamiltonian in terms of a BdG Hamiltonian (I recall that this is what Furusaki and Ludwig did) which makes the PH operation "more obvious".

 

[ComradeVVA]

Thanks Kirjava! I am currently looking at a paper, http://arxiv.org/abs/1309.0878, and have always had an interested for the more abstract aspects of the topic.

Your answer makes sense to me for R_2,3, but not for R_4, whose class (AII) has no particle-hole symmetry and only time reversal, yet Kitaev's case has both T and QT.

 

[Kirjava]

Let me address a possible misconception first: the correspondence between AZ symmetry classes and the classifying spaces is "dimension dependant", so it doesn't quite make sense to equate R_4 with the symmetry class AII.

Regarding your question: you're right that a PH symmetry is not present in class AII, but Kitaev is actually working in a "redundantly large" particle-hole (vector) space from the begining, so the PH symmetry which I told you is simply complex conjugation is more of a redundancy in the description of particle-non-conserving Hamiltonians than a physical symmetry. The Q symmetry he introduces is a way of talking about particle-conserving (i.e. single-particle) Hamiltonians using the same formalism.

 

[ComradeVVA]

Regarding your first point, I apologize for not clarifying: I'm focused on d=0. Figured if I understand that, the periodicity will take me to any other d.

I think I now understand better the connection between Kitaev's iA Hamiltonian in Majorana basis, Kitaev's X Hamiltonian in regular basis, and everybody else's first-quantized Hamiltonians H. I've used Furusaki's paper as sort of a conversion tool. Attached is an attempt to understand Kitaev's point of view compared to Morimoto/Furusaki's. Feel free to correct/comment.