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After thinking about whether this discussion should go to any or all of topological phases of matter, topological states of matter, topological order, topological insulators (quite a mess of terminology!), let me give it it’s own stand-alone entry hereby.
Similarly, I will finally create a corresponding stand-alone entry for K-theory classification of D-brane charge
Eventually I’ll write up a decent Idea-section for this entry, but for the moment I have a question to the experts reading here (if any):
Besides the generic non-spatial CPT-symmetries that make the 10-fold way, I suppose a crystalline insulator could exhibit further non-spatial internal symmetries , no?
I imagine in general there may be a discrete group $G_{int}$ of “internal symmetries” acting on the internal degrees of freedom of the electrons at one lattice site.
For example, if any spin-orbit coupling etc. may strictly de disregarded, then there ought to be an extra non-spatial $G_{int} = \mathbb{Z}/2$ “spin reversal symmetry” enjoyed by the material, and we might ask the K-theory classification to respect that.
But generally it feels plausible that larger finite groups $G_{int}$ could act as non-spatial symmetries on lattice sites, such that if we ask for classifications respecting this extra non-spatial symmetry, then we should include such a factor of $G_{int}$ in the equivariance group for the K-theory.
Has this been discussed anywhere?
In short, I am probably asking very simply if the combination of
“symmetry protected topological order”
“K-theory classification of topological phases”
has been discussed in a substantial way anywhere?
It sounds like the answer should trivially be “Yes, of course!” but maybe there is a little disconnect between the two communities who subscribe to these two terms, respectively.
For instance arXiv:1906.02892 speaks as if these are two mutually exclusive concepts (which I wouldn’t think they are):
The question now is this: is there a general mathematical framework for the classification of SPT phases in the same way as there is K-theory for topological insulators?
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