Counterexamples to the bulk-boundary correspondence (topological insulators)

[Kirjava]

In the literature on topological insulators and superconductors the 'bulk-boundary correspondence' features quite heavily. One version of this conjecture says roughly: "At an interface between two materials belonging to the same symmetry class with bulk invariants n and m, precisely |n-m| gapless edge modes will appear". Are there any known counterexamples to this statement when the invariants are of the usual non-interacting Bloch band type? (specifically I have in mind the invariants appearing in the "periodic table" of T.I.s/T.S.Cs, see 0901.2696 and 0912.2157). As far as I know no comprehensive proof of the statement exists, although considerable supporting evidence has been found in a number of special cases.

As some extra motivation, suppose that there are new bulk invariants waiting to be found protected by symmetries falling outside the usual classification schemes (e.g. the recently proposed topological crystalline insulators protected by point group symmetries). Are there any known reasons to be confident that the bulk-boundary correspondence will continue to hold in these cases?

 

[Physics Monkey]

If I understand your question correctly, I think its basically settled at the non-interacting level. In some cases we have a very general reason for believing the bulk-edge correspondence in complete generality. For example, the so-called gravitational anomaly in 1+1d, which just corresponds to the chiral conduction of heat, essentially rigorously shows that at least n-m modes must survive e.g. chiral p-wave scs. The argument in the other direction is simply that if we allow every possible coupling then every mode that doesn't have to be there will disappear.

Still, there are all kinds of subtleties. For example, the symmetry which protects the edge modes could spontaneously break at the surface. Also, as I'm sure you're aware, the edge/surface modes don't always come in integers, so sometimes 2 is the same as none e.g. in 3d TI.

Finally, there are plenty of reasons to believe that the bulk edge correspondence doesn't hold for things like topological crystalline insulators. This is because boundaries in these and other models e.g. those with inversion typically have to break the relevant symmetry and hence have nothing to protect them. This is not to say that TCI will never have edge states, since all kinds of materials just happen to have such states, but I do think it implies that PROTECTED edge states are not really a feature of such phases.

 

[Kirjava]

There appear to be a number of different approaches to characterizing the bulk of a topological insulator (some of which may avoid this difficulty), but the simplest non-interacting methods are formulated in the language of Bloch wavefunctions and Hamiltonian. An edge or interface obviously breaks the translational symmetry needed to make sense of these, so by your comment on TCIs one might think that the standard topological invariants (chern and spin-chern numbers etc.) are similarly meaningless, no?

Can I follow up on your second sentence as well? "in some cases we have a very general reason..." seems to confirm my suspicion that there is still no universal understanding of the correspondence which can be conveyed in a sentence or two. One initially promosing candidate which I believed for a while says something like "bulk invariants cannot change without closing the gap, so at an edge between two materials with different values for this quantity the gap must close". The problem with this is that there is no need for the transition to be even remotely continuous (e.g. Tight binding models where the interface is a single lattice plane). I'm quite surprised that I ever read this in a serious paper actually (I think it might have been Hasan's review). Or maybe there's more to it than meets the eye?

 

[Physics Monkey]

The difference between a TI and TCI is that in the former the relevant symmetry is not translation or inversion but time reversal. So while it is true that the edge breaks various symmetries, a TI will remain gapless so long as time reversal is not broken.

Your point about the use of Bloch states is a good one, but ultimately one should think about these devices as calculational tools instead of fundamental frameworks. There are now a vareity of ways to define and compute topological indices with bulk disorder e.g. non-commutative geometry, scattering matrix formalism, C* algebras, Green's functions, boundary condition twists, and so on ...

The story about disorder is an especially old one in the context of quantum Hall effect. There one has a picture of edge states flowing around any disorder you might care to place on the edge.

For my money, one of the better ways to think about such states is in terms of the low energy effective theory. For quantum Hall states you have the Hall conductivity, for 3d TI you have the theta term, and so on. These low energy descriptions give a sharp experimental distinction between the two phases and they can often be used to show that edge states must exist.

A very important issue related to this is the problem of interactions. For example, the non-interacting classification gives Z for a certain kind of superconductor in 1d, but Kitaev and Fidkowski showed that interactions reduce this to Z8. But at the same time, part of the reason why we were unsure in this case, even before Kitaev and Fidkowski, was that we didn't have a simple low energy theory that could distinguish different phases.

The problem with the statement in your second paragraph is that its false. There are kinds of interacting phases with different topological structures which are not guaranteed to have protected edge states. So it may be roughly true for certain kinds of simple non-interacting phases, but more generally, determining when boundary states MUST exist at the interface of two phases is much more complicated. At the moment we only have partial answers.

 

[Kirjava]

Thanks for the attentive replies as usual. I understand that once interactions are added things become less clear, and there are (many?) known interacting cases where the BBC fails. Nevertheless I think it's still an interesting question to ask whether it holds in all the known non-trivial non-interacting phases.

The difference between a TI and TCI is that in the former the relevant symmetry is not translation or inversion but time reversal. So while it is true that the edge breaks various symmetries, a TI will remain gapless so long as time reversal is not broken.

While this certainly seems to be the message emerging from studies of the IQHE & QSHE, I don't know how this squares with the idea that the usual topological invariants delineate classes of systems (Hamiltonians) related by continuous deformation. For example, the TKNN number can be understood as a class in $\pi_2$ of some classifying space (a Grassmanian in this case I think), but if we drop the requirement of translational symmetry the classes are given by $\pi_0$ instead, which is at least different if not empty. This suggests to me that preserving the gap and generic symmetries (none for IQHE) is not enough, and there must be some extra requirement so that the disordered system "remembers" its translational invariance. I suppose the other techniques you refer to for computing invariants must incorporate such a condition.

 

[Physics Monkey]

Thanks for the attentive replies as usual. I understand that once interactions are added things become less clear, and there are (many?) known interacting cases where the BBC fails. Nevertheless I think it's still an interesting question to ask whether it holds in all the known non-trivial non-interacting phases.

I'm not sure exactly how I would characterize the failure. In a sense, there is still a kind of correspondence in that if the edge modes can be removed by turning on interactions then the bulk phases are often also smoothly connected in the presence of interactions. This is what Kitaev and Fidkowski showed in a particular case. But you are right more generally in that there are topological phases without edge states and topological phases with edge states that nevertheless remain distinct even when the edge states can be gapped out at an interface.

While this certainly seems to be the message emerging from studies of the IQHE & QSHE, I don't know how this squares with the idea that the usual topological invariants delineate classes of systems (Hamiltonians) related by continuous deformation. For example, the TKNN number can be understood as a class in $\pi_2$ of some classifying space (a Grassmanian in this case I think), but if we drop the requirement of translational symmetry the classes are given by $\pi_0$ instead, which is at least different if not empty. This suggests to me that preserving the gap and generic symmetries (none for IQHE) is not enough, and there must be some extra requirement so that the disordered system "remembers" its translational invariance. I suppose the other techniques you refer to for computing invariants must incorporate such a condition.

If I understand your objection correctly, the key point is locality. Even without translation invariance the quantum Hall case is still local in 2d. So the Hamiltonian cannot be taken to be a generic 2^N by 2^N matrix (N electron sites, say). It must still respect locality so in that sense it remains quite unlike a lower dimensional system. Another point of concern may be that one still wants some kind of thermodynamic limit. For example, if the disorder is so strong that the system breaks up into disconnected chunks, then that is clearly beyond the validity of the classification scheme and its claimed robustness to disorder.