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

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Referee this paper: arXiv:1410.8477 by Xiao-Gang Wen

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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 $\text{iTO}_L^1=\text{iTO}_L^2=\text{iTO}_L^4=\text{iTO}_L^6=0$, $\text{iTO}_L^3=\mathbb{Z}$, $\text{iTO}_L^5=\mathbb{Z}_2$, $\text{iTO}_L^7=2\mathbb{Z}$.

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
paper authored Oct 30, 2014 to cond-mat
edited Nov 15, 2014

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.

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