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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 12th 2020
   • (edited Feb 12th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAfter 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]]_ <a href="https://ncatlab.org/nlab/revision/K-theory+classification+of+topological+phases+of+matter/1">v1</a>, <a href="https://ncatlab.org/nlab/show/K-theory+classification+of+topological+phases+of+matter">current</a>

   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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2022
   • (edited May 7th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexEventually 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn short, I am probably asking very simply if the combination of 1. "symmetry protected topological order" 1. "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](https://arxiv.org/abs/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?

   In short, I am probably asking very simply if the combination of

   1. “symmetry protected topological order”

   2. “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?