Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Construction of bosonic symmetry-protected-trivial states and their topological terms via $G\times SO(\infty)$ non-linear $σ$-models

Originality
+ 3 - 0
Accuracy
+ 1 - 0
Score
2.32
4894 views
Referee this paper: arXiv:1410.8477 by Xiao-Gang Wen

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

(Is this your paper?)


Summary by author Xiao-Gang Wen:

It has been shown that bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via non-linear $\sigma$-models (NL$\sigma$Ms) of the symmetry group $G$ with various topological terms.  Those SPT phases (called the pure SPT phases) can be classified by group cohomology ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$.  But there are also SPT phases beyond ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$ with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases).  Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. This paper shows that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. 

More precisely, the paper  shows that both the pure and the mixed SPT phases can be realized by $G\times SO(\infty)$ NL$\sigma$Ms with various topological terms. Through the group cohomology ${\cal H}^d[G\times SO(\infty),\mathbb{R}/\mathbb{Z}]$, one finds that the set of constructed SPT phases  in $d$-dimensional space-time  are given by

$E^d(G)\rtimes \oplus_{k=1}^{d-1} {\cal
H}^k(G,\text{iTO}_L^{d-k})\oplus {\cal H}^d(G,\mathbb{R}/\mathbb{Z}) $.

Here $\text{iTO}_L^d$ is the set of the topologically-ordered phases in $d$-dimensional space-time that have no topological excitations. One finds that $\text{iTO}_L^1=\text{iTO}_L^2=\text{iTO}_L^4=\text{iTO}_L^6=0$, $\text{iTO}_L^3=\mathbb{Z}$, $\text{iTO}_L^5=\mathbb{Z}_2$, $\text{iTO}_L^7=2\mathbb{Z}$. 

SPT phases

Black = the pure SPT phases described by ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$. Blue = the mixed SPT phases described by $\oplus_{k=1}^{d-1} {\cal
H}^k(G,\text{iTO}_L^{d-k})$ but beyond ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$. Red = the extra mixed SPT phases described by $E^d(G)$.

It may be possible that $G\times SO(\infty)$ NL$\sigma$Ms with various topological terms can realize all the pure and mixed SPT phases that can be defined on arbitrary space-time manifolds

The NL$\sigma$M construction also gives the topological terms that fully characterize the corresponding SPT and iTO phases. Through several examples, the paper shows how can the universal physical properties of SPT phases be obtained from those topological terms.

summarized by dimension10
paper authored Oct 29, 2014 to cond-mat by Xiao-Gang Wen
  • [ revision history ]
    edited Nov 15, 2014 by dimension10

    Hi Xiao-Gang. I apologize for not reading your paper yet, but I hope you don't mind some naive questions. Do you assume smoothness of spacetime? Also it seems that your set of phases is the E_2 page of the Atiyah-Hirzebruch spectral sequence. This spectral sequence does not always degenerate so quickly (although perhaps it does for all the symmetry groups you consider), so there will be some collapsing of the classification for more complicated symmetries (and especially with fermions). Is this right?

    Ryan: I need to learn Atiyah-Hirzebruch spectral sequence. The $\oplus_{k=1}^{d-1} {\cal
    H}^k(G,\text{iTO}_L^{d-k})$ part of my result is the E_2 page. I also have the $E^d(G)$ part which is for extra things (the red entries in the above table), that are beyond the $E_2$ page. Kapustin has pointed this out to me. Do you know an easy ref for  Atiyah-Hirzebruch spectral sequence?

    Hi Ryan: If one needs to go beyond the $E_2$ page, do you have more SPT states or less SPT states? I have an arguement in the paper that we at least have $\oplus_{k=1}^{d-1} {\cal
    H}^k(G,\text{iTO}_L^{d-k})$. See discussions below eq. 104.

    There will in general be less (never more), since at each page we are passing to the cohomology of the differentials on that page. It's just like how the dimension of the space of chains is an upper bound on the Betti number. And unfortunately I have still not yet found a good reference.
    @JiaYiyang Ok I've moved the summary and its comments to the submission itself.
    I have an arguement in the paper that we at least have $\oplus_{k=1}^{d-1} {\cal H}^k(G,\text{iTO}_L^{d-k})$ for the SPT states. If $\oplus_{k=1}^{d-1} {\cal H}^k(G,\text{iTO}_L^{d-k})$ is the E_2 page of the Atiyah-Hirzebruch spectral sequence, then it seems suggest that we at most have $\oplus_{k=1}^{d-1} {\cal H}^k(G,\text{iTO}_L^{d-k})$ for the SPT states. Does this means that $\oplus_{k=1}^{d-1} {\cal H}^k(G,\text{iTO}_L^{d-k})$ is an one-to-one description of SPT states? In the paper, I found more SPT states than $\oplus_{k=1}^{d-1} {\cal H}^k(G,\text{iTO}_L^{d-k})$. But I could miss something and maybe there are no more.

    Your Review:

    Please use reviews only to (at least partly) review submissions. To comment, discuss, or ask for clarification, leave a comment instead.
    To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
    Please consult the FAQ for as to how to format your post.
    This is the review box; if you want to write a comment instead, please use the 'add comment' button.
    Live preview (may slow down editor)   Preview
    Your name to display (optional):
    Privacy: Your email address will only be used for sending these notifications.
    Anti-spam verification:
    If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
    p$\hbar$ysics$\varnothing$verflow
    Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
    Please complete the anti-spam verification




    user contributions licensed under cc by-sa 3.0 with attribution required

    Your rights
    ...