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

TL;DR

The authors unify the construction and classification of bosonic SPT and iTO states using G×SO(∞) nonlinear sigma models, showing that L-type phases are captured by ${\cal H}^d(G\times SO, {\mathbb R}/{\mathbb Z})$ and organize them into pure and mixed sectors via ${\cal H}^d(G,{\mathbb R}/{\mathbb Z})$, $\bigoplus_{k} H^k(BG, {\rm iTO}_L^{d-k})$, and the extra group $E^d(G)$. They introduce universal fixed-point invariants $W^d_{top}$, derived from cocycles, which fully characterize SPT and iTO phases under symmetry twists and gravitational couplings, and provide a detailed decomposition and realizability analysis (L-type) for simple symmetry groups such as U(1), Z_n, and time-reversal variants. The work connects the group-cohomology description to mixed gauge-gravity anomalies, cobordism, and the lattice NLσM framework, offering explicit constructions, dimension-reduction interpretations, and a roadmap for probing topological properties through symmetry twists and gauging. Collectively, the paper delivers a comprehensive, algebraically organized, and physically interpretable classification and realization scheme for a broad class of bosonic topological phases with and without symmetry.

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 $σ$-models (NL$σ$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$σ$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.

Figures (10)

• Figure 1: (a) a loop creation. (b) a loop annihilation. (c) a line reconnection.
• Figure 2: (Color online) (a) a point is split into three points. (b) a surface $N^2$ is split into three surfaces $N^2_1$, $N^2_2$, $N^2_3$.
• Figure 3: (Color online) Loop annihilation: (a) as we shrink the black circle to a point, the black line sweeps across the intersection of red and blue line once. This means that $N^2_1$, $N^2_2$, $N^2_3$ intersect once in the loop annihilation/creation process. Line reconnection: as we deform the black lines in process (b), the black lines sweep across the intersection of red and blue lines once. But in process (c), no line sweeps across the intersection of the other two lines. This means that $N^2_1$, $N^2_2$, $N^2_3$ intersect once in the line reconnection process.
• Figure 4: (Color online) (a) a $Z_2$ symmetry twist on a torus. (c) the $Z_2$ symmetry twist obtained from (a) by double Dehn twist. (a$\to$b$\to$c) contains a line reconnection.
• Figure 5: (Color online) Deformation of the $Z_2$-twist lines and two reconnection moves, plus an exchange of two defects and a $360^\circ$ rotation of one of the defects, change the configuration (a) back to itself. Note that from (a) to (b) we exchange the two defects, and from (d) to (e) we rotate of one of the defect by $360^\circ$. The combination of those moves do not generate any phase, since the number of the reconnection move is even.