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After having dug through the literature, I am growing convinced that it is unknown why topological phases of matter are classified by (twisted equivariant) K-theory.
The only actual argument that seems to exist is still the original two short sentences of vague hand-waving that Kitaev offered (top left of p. 4):
We generally augment by a trivial system, i.e., a set of local, disjoint modes, like inner atomic shells. This corresponds to adding an extra flat band on an insulator.
Apart from the question whether the second sentence really implies a classification by K-theory, the first sentence does not even seem to be a proper physics argument, which instead would require invoking some actual facts in Bloch-Floquet theory.
Here the state of the art is Panati’s theorem (arXiv:math-ph/0601034, arXiv:1108.5651, arXiv:1601.02906). Unfortunately, Panati says “Bloch bundle” for what really is the “valence bundle” and this has tricked some authors (e.g. those of p. 5 in arXiv1310.0255 as well as the author of my last Physics.SE post, which I have since deleted) into thinking that Panati’s result is what is missing in Kitaev’s argument. But it’s not.
Worse, the probably second-most cited article on the matter after Kitaev’s openly says (p. 57) that:
Although [K-theory] is used in the condensed matter literature, it is not clear to us that it is well motivated.
While this state of affairs is from over 8 years ago, I haven’t seen any more recent articles that would even attempt to improve on this.
That said, I am beginning to feel that I see how it should work:
First, what are we actually asking for?
I think we are really asking for classification of charged vacuum ground states of the free electron/positron field in the background of a scalar potential (periodic, in our case, reflecting the crystals’ atomic sites); the charge being the number of bands below the given gap energy.
Such vacua have been computed, long ago, in
The result is that these vacua are characterized by the projection of the “dressed” electron states on the “bare” electron states, with the vacuum’s charge being the dimension of this operators cokernel minus kernel.
A little later, the authors of
observe (surveyed on their p. 364) that, as this might suggest, the characteristic operator is Fredholm, and the vacuum charge is its index.
So that’s it, I think now: Under Bloch-Floquet this gives us a bundle of vacua over the Brillouin zone classified by a family of Fredholm operators. The homotopy class of that family is the expected K-theory class of the topological phase.
Cool piece of literature archaeology there, Urs! So, to sum up: the direct claim that K-theory is what classifies topological phases, as attested in the literature, is based on citations that are more guesswork and hand-waving—but an actual solid argument can be pieced together from existing old papers, though it isn’t cited even indirectly. Is that about right?
(On a personal note, I had the pleasure of knowing Angas Hurst briefly in his old age, unfortunately not at a time when I was mathematically mature enough to have a really serious discussion with him.)
It takes some more work to turn the observation in the second part of #2 into a proof: One should show that (a) the argument given there for the full Hamiltonian applies to each Bloch Hamiltonian, (b) that the resulting family of Fredholm operators is continuous over the Brillouin torus and (c) that homotopy classes of such Fredholm families do correspond to physical deformation classes.
But at least it is a concrete and plausible path to a proof.
What is fascinating about it is that it would finally explain what the virtual bundles in the K-theory mean physically (which is a point not even touched upon in Kitaev’s argument for stable equivalence): Namely, we see that in a proper relativistic discussion of energy bands, the valence electron bundle may have a positronic contribution, and that positronic valence bundle contributes with negative sign to the K-theory class. In other words, the Grothendieck group completion that takes the stable equivalence classes envisioned by Kitaev to actual K-theory classes physically corresponds to electron/positron annihilation.
We are going to write this up in as far as it goes without entering into a genuine solid state physics research project, because we need to be focusing on something else. But if an energetic condensed matter theorist sees this here and feels like joining in, drop me a note.
added more of the original articles:
Michael Stone, Ching-Kai Chiu, Abhishek Roy, Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock, Journal of Physics A: Mathematical and Theoretical, 44 4 (2010) 045001 $[$doi:10.1088/1751-8113/44/4/045001$]$
Gilles Abramovici, Pavel Kalugin, Clifford modules and symmetries of topological insulators, International Journal of Geometric Methods in Modern PhysicsVol. 09 03 (2012) 1250023 $[$arXiv:1101.1054, doi:10.1142/S0219887812500235$]$
added this item at the top:
Precursor discussion phrased in terms of random matrix theory instead of K-theory:
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