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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeOct 10th 2013
• (edited Oct 10th 2013)

Xiao-Gang Wen has started an entry symmetry protected trivial order.

He mentions ’group super-cohomology theory’ as describing fermionic SPT orders. Given our super-slick account of group cohomology, do we just change the ambient $\infty$-topos to Super $\infty$ Grpd?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 10th 2013

Thanks for highlighting, I would have missed it otherwise.

Yes, if this means (as it seems it does) group cohomology of super-Lie groups, then, yes, this is just mapping spaces in $Super\infty Grpd$.

Right now we are talking about just such super Lie group cohomology at our Super Gerbes meeting. And that’s why right now I have to run and quit reading here. But I’ll try to get back to this later this evening. Thanks again for the heads up.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeOct 10th 2013
• (edited Oct 10th 2013)

Started a page for Xiao-Gang Wen, who is now at the Perimeter Institute. The description there mentions ’condensed matter’ theory. We could do with an entry on that. Someone at Princeton gives it a go here. So solid-state physics is now seen as a branch of condensed matter physics.

EDIT: Oh, that’s just taken from wikipedia Condensed matter physics.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 10th 2013

We have had a stub solid state physics for some time. I have added more redirects.

• CommentRowNumber5.
• CommentAuthorxgwen
• CommentTimeOct 11th 2013
Group super-cohomology theory is a term that Gu and I invented. It is not the group cohomology for supergroup. It is a "group cohomology" with anticommuting coefficient. The usual group cohomology with U(1) coefficient is denoted as H^d[G,U(1)].
In group super-cohomology we replace U(1) by something that contain anticommuting Grassman numbers. We really want to know is there such kind of group cohomology theory in math (maybe under a different name). I would like to thank Urs for editting the SPT entry.
• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 11th 2013

Thanks for the information. I would expect that also your definition of cohomology with super-geometric coefficients is still given by maps in the higher supergeometric topos.

Could you point me to the precise page of an article where the group super-cohomology in your sense is defined? Thanks!

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013

In this paper, we introduce a (special) group super-cohomology theory

See Appendix C on p. 35.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 7th 2015
• (edited Oct 7th 2015)

I have finally begun to cross-linke symmetry protected topological order with higher dimensional WZW model, due to the article

• Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013) arXiv:1106.4772; A short version in Science 338, 1604-1606 (2012) pdf

which argues that the bosonic SPT phases are described just by such higher WZW models.

This needs to be expanded on.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMay 7th 2022

I have re-written and expanded the Idea-section of this entry.

Also, I have changed the title from “… trivial order” to “… topological phase”, to be less confusing.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMay 7th 2022

also I am polishing up some of the old previous material here (in large part by X.-G. Wen himself), such as properly bringing out this reference item:

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMay 7th 2022
• (edited May 7th 2022)

added the reference for the “first” (and only) example that the entry has been mentioning:

• Duncan Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the $O(3)$ nonlinear sigma model, Physics Letters A 93 9 (1983) 464-468 $[$doi:10.1016/0375-9601(83)90631-X $]$
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMay 7th 2022
• (edited May 7th 2022)

I have trouble spotting the argument (or proof, if that’s the right word) for the claim about the classification of SPTs “by group cohomology” – it does not seem to be in the two articles that are usually cited for this claim.

Now I see that an argument is spelled out in Sec. V of:

But this uses a peculiar concrete lattice model and, it seems to me, some more assumptions.

In any case, I have added this pointer now.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeMay 9th 2022

I have expanded a fair bit all three items in the remark on terminology conventions (here).

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeMay 11th 2022
• (edited May 11th 2022)

added mentioning of the notion of “symmetry enriched topological” (SET) phases (right at the beginning, here)

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeMay 11th 2022

I have adjusted the content of the section “Classification” a little (here) and then I have added the following cautionary quote from a more recent article (BBCW 19):

Although a remarkable amount of progress has been made on these deeply interrelated topics, a completely general understanding is lacking, and many questions remain. For example, although there are many partial results, the current understanding of fractionalization of quantum numbers, along with the classification and characterization of SETs is incomplete.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeMay 11th 2022

also added (still here) from Wang & Senthil 14, p. 1

this classification is now known not to be complete

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeMay 13th 2022

I have added in the section on Classification of internal-SPT phases (here) some indication of the basic proposal due to

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeMay 13th 2022

Have appended to this (here) a couple of paragraphs on how inspection of TE-K theory seems to resolve the issue.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeMay 15th 2022

and took from it a further quote (now here) on how the classification of SPTs remains an open problem.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeMay 15th 2022

added a table (here) showing organization and examples of types of symmetries to which the discussion applies (external spatial, external CT, internal/on-site)

will add this also to related entries, such as at internal symmetry

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeMay 16th 2022

I have re-worded a fair bit the list of types of protecting symmetries here, bringing the prime example of internal $Spin(3)$-symmetry more to the forefront

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeMay 16th 2022

I am wondering now about the followint CMT- Question:

What’s a plausible way to realize/find (approximate) cyclic-group ($\mathbb{Z}/n$) on-site symmetry protection in actual crystals/materials?

More specifically:

What’s a plausible way to realize in actual materials a cyclic group on-site symmetry protection where $\mathbb{Z}/n \subset SU(2) \simeq Spin(3)$ is a maximal subgroup of the on-site electron spin-rotation which remains unbroken?

Is this at all something one expects to ever see in the laboratory?

And if the answer should be “Yes” then how about the other finite subgroups of SU(2)?

Could there be an ADE-classification of SPT mechanisms in realistic materials where the protecting symmetry group is a finte subgroup of on-site spin rotation of electrons?

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeMay 17th 2022

I have now forwarded this question to Physics.SE: here

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTime1 day ago

added pointer to today’s

• Zhihuang Luo, Wenzhao Zhang, Xinfang Nie, Dawei Lu, Observation of a symmetry-protected topological phase in external magnetic fields [arXiv2208.05357]