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

TL;DR

This paper develops a unified framework using $G imes SO( olinebreak ext{infinity})$ non-linear sigma models to classify both pure and mixed bosonic SPT phases via an extended group cohomology approach, revealing new topological invariants.

Contribution

It introduces a novel classification scheme for SPT phases that includes beyond-group-cohomology states using $G imes SO( ext{infinity})$ NL$\sigma$Ms and identifies their topological invariants.

Findings

Unified classification of pure and mixed SPT phases.

Explicit topological invariants for different SPT and iTO phases.

Examples demonstrating physical properties derived from invariants.

Abstract

It has been shown that the L-type 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 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. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show 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 ${\cal H}^d[G\times SO(\infty),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed L-type SPT phases in $d$-dimensional space-time are classified 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}) $ where $G$ may contain time-reversal. Here $\text{iTO}_L^d$ is the set of the L-type topologically-ordered phases in $d$-dimensional space-time that have no topological excitations, and one has $\text{iTO}_L^1=\text{iTO}_L^2=\text{iTO}_L^4=\text{iTO}_L^6=0$, $\text{iTO}_L^3=\mathbb{Z}$, $\text{iTO}_L^5=\mathbb{Z}_2$, $\text{iTO}_L^7=2\mathbb{Z}$. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how can the universal physical properties of SPT phases be obtained from those topological invariants.