Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2020

    a bare list of references, to be !include-ed into the references lists of relevant entries

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2022

    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

    • M. Klaus, G. Scharf, The regular external field problem in quantum electrodynamics, Helvetica Physica Acta 50 (1977) (doi:10.5169/seals-114890, pdf)

    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

    • A. L. Carey, C. A. Hurst, D. M. O’Brien, Automorphisms of the canonical anticommutation relations and index theory, Journal of Functional Analysis 48 3 (1982) 360-393 (doi:10.1016/0022-1236(82)90092-1)

    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.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 3rd 2022

    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.)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2022
    • (edited Apr 4th 2022)

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2022

    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]]

    diff, v6, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2022

    added this item at the top:


    Precursor discussion phrased in terms of random matrix theory instead of K-theory:


    diff, v7, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)