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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:
am adding more precursor articles which effectively were looking at the K-theory classification already (not quite, but essentially as far as Kitaev09 did, too), without calling it by that name:
added pointer to:
also pointer to:
added pointer to:
added pointer to:
added pointer to:
added pointer to:
I’m currently reading anyonic topological order in TED K-theory and am having some difficulty understanding the meaning of fact 2.3. To be transparent I’m not trained in the necessary physical theory to understand the jargon, and thus some of the listed resources are a bit unreadable for me.
A brief (mathy) description of each of the $P_{(-)}$,$U$, and $\gamma$ and their physical significance would be very helpful!
It’s stated that $\gamma:H\oplus H\to H\oplus H$ is just an isomorphism exchanging the factors of the Dirac particle state space, but this means my understanding of the $P_{(-)}$’s as projections $H\oplus H \to H$ is incorrect since the composition $\gamma\circ P_{(-)}$ wouldn’t quite make sense without some further stipulation (naively something like extension to the second factor by zero).
I’m not even sure where to begin with the unitary operator $U$ since I’m not sure what the meanings of “bare” and “dressed” are in any physical or mathematical sense, nor how we can use these to define $U$ as some unitary operator on $H\oplus H$ with the property that it acts on the corresponding fermionic Fock space (exterior algebra) in a certain way.
Wouldn’t you typically use the action of an operator on generators to define its action on the exterior algebra and not the other way around? I mean, I suppose that if you defined the action of something on the exterior algebra you must have necessarily defined it on generators and so you get what you’re after anyways, but I digress (and am likely making clear that I have no clue what’s going on here).
Thanks for the help, and I’m sorry that my confusion doesn’t lend itself to a more explicit set of questions!
The mathematical statement is (as cited) that from p. 364 of
They write the Fredholm operator as $P_+ U P_+$ (we display it composed with the $\gamma$-isomorphism just in order to regard it equivalently as a graded Fredholm operator for better use in (Karoubi’s) K-theory later on).
A “projection operator” in linear algebra and functional analysis is understood as an idempotent endomorphism (in contrast to the “product projections” in category theory). Here this means that:
$\array{ \mathllap{P_+ \,\colon\,} \mathscr{H} \oplus \mathscr{H} &\longrightarrow& \mathscr{H} \oplus \mathscr{H} \\ (\psi_+, \psi_-) &\mapsto& (\psi_+, 0) }$and analogously for $P_-$.
The physics jargon of “dressed” vacua refers to change of notion of elementary “particle” or “excitation” as some interaction is turned on: Since we are concerned with the Dirac field in the background Coulomb field of the nuclei in the crystal lattice, the usual electron/positron states which are eigenstates of the free Dirac equation (as discussed in standard references such as Thaller 1992) are no longer energy eigenstates of the Dirac equation with that Coulomb potential term included. Nevertheless, since they form a linear basis of the full Hilbert space, the actual eigenstates in the presence of the Coulomb potential are expressible as linear combinations of the free ones.
This transformation from states in the trivial vacuum to states in the actual Coulomb vacuum is the “dressing” denoted”W” (which is the same as the “$U$” from Carey et al., above) in (3.9) of
Section 3 “The dressed electron-positron states” in:
These authors spell out their argument in much detail.
In particular, $P_\pm$ has the phyiscal interpretation of projecting (in the above sense) onto the sub-Hilbert spaces of dressed electron or positron states — equation (3.2) in Klaus & Scharf — which we may think of as the actual physical electron/positron states in the given background Coulomb potential.
The achievement of Klaus & Scharf is to consider and solve the evident but rarely discussed question of what happens to the Dirac vacuum when a background Coulomb field is turned on.
The achievement of Carey et al. is to provide a streamlined mathematical summary of their result — which seems to be what you are asking for.
All we add on this point is the observation that this finally seems to explain why free crystalline phases are classified by Bloch families of Fredholm operators and hence by the K-theory of the Brillouin torus.
This is exactly the response I was after! Thank you!
Hi Urs, I have another question regarding anyonic topological order in TED K-theory (if there’s a better place to post questions for this paper let me know! I wasn’t sure which discussion page would be most adequate).
In examples 2.15 and 2.16, we see how topological insulating phases with some given symmetry protections should be classified by $KO^{0},KO^{4},$ and $KU^{}$. In both of these examples there are displayed tables of the respective K-groups of spheres for each case.
I can see in example 2.16 that in dimensions $d=1,2$ we have $KU^{0}(\mathbb{T}^{d})\cong \KU^{0}(S^{d})$, so in particular these entries in the table say that in 1 spatial dimension we have no non-trivial unprotected phases and that in 2 spatial dimensions we have non-trivial unprotected phases classified by integers (integer quantum Hall); but certainly $KU^{0}(\mathbb{T}^{3}) \cong KU^{0}((S^{1})^{\times 3})\ncong KU^{0}(S^3)$ and similarly for the other entries (unless I’ve made a mistake).
How are these tables related to the classification of the topological phases by the K-theory of Brillouin tori in each example? I’ve seen that the classification of class A insulators is exactly the table presented in example 2.16, but aren’t there non-trivial $KU^{0}(\mathbb{T}^{d})$ for all $d\geq 2$ by employing
$KU^{-n}(X\times Y) \cong KU^{-n}(X\wedge Y)\oplus KU^{-n}(X)\oplus KU^{-n}(Y)$together with the suspension isomorphism?
Yes, that sounds right. (The last iso should be in reduced K-theory, to be precise.)
If I understand you well, this is not so much a question about what we have written than about what is written elsewhere.
Yes I suppose that’s right! Ultimately I wanted to make sure that I understood what these tables of K-groups of spheres were trying to illustrate. At first I thought somehow your formalism would recover them which was my cause for confusion as this isn’t the case, but it looks like they make plain the differences between your classification and the previous attempts at a classification by others. If I understand correctly then, your formalism posits a much richer structure for unprotected insulators in $d$ > $2$ than what was previously included in the folklore, very cool!
No, wait: at that point we are reviewing the usual K-theory classification of free topological phases.
The new step we take is later in section 3, where we posit that the interacting topological phases are classified by the (twisted, equivariant) K-theory not of the plain Brillouin torus, but of its configuration space of points.
Okay, I’m still a bit confused then. In the usual classification for class A insulators we only see non-trivial phases in even dimensions corresponding to the Chern insulators, these are each classified by a copy of $\mathbb{Z}$. If $KU^{0}(\mathbb{T}^{d})$ is non-trivial for all $d\geq 2$, how do we recover the usual result?
It seems somehow that the K-groups for the tori are meant to be related to the K-groups of spheres, which do produce $\mathbb{Z}$ in even dimension and 0 in odd, but I don’t see how in $d\gt 2$.
There’s a later reference under figure 7 to Twisted equivariant matter where the authors decompose the torus into a wedge sum of spheres, but in d=3 for example we see that $KU^{0}(\mathbb{T}^{d})\cong \mathbb{Z}^{\oplus 3}$ coming from the three copies of $S^2$ in the decomposition, but this doesn’t recover the previous result that there are no Chern insulators in dimension 3. Unless this isn’t the previous result at all and I’ve misunderstood the “periodic tables” floating around the literature.
Yes, that’s why I suggested in #19 that your issue is with the rest of the literature, not with what we wrote.
Beware that the topological CMT literature is most vague and statements have to be taken with grains of salt. People who speak about periodicity as it seems you have seen them speak will secretly (whether consciously or not) refer to evaluation of K-theory on the fundamental (top-dimensional) classes.
One account of Chern insulators which is more careful (on the theory side) is Mathai & Thiang 2017b (which is our reference [MT16]).
(They speak in the generality of Chern-semimetals in 3d, but their discussion subsumes the Chern-insulator invariants. Also, they consider only ordinary cohomology instead of K-theory, but for the Chern-case this makes essentially no difference.)
You can see that they do pick up the Chern-numbers associated to 2-cycles inside the 3-torus which you are wondering about. They refer to them as “weak Chern insulater invariants” (search the document for the string $\;\;\;$ "weak" Chern
$\;\;\;$ with the quotation marks included!), such as in item (3) on p. 11.
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