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

It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear $\sigma$ models $\left(\mathrm{NL}\sigma \mathrm{Ms}\right)$ of the symmetry group $G$ with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology ${\mathcal{H}}^d(G,\mathbb{R}/\mathbb{Z})$. 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 $G\times SO\left(\infty \right) \mathrm{NL}\sigma \mathrm{Ms}$ with various topological terms. Through the group cohomology ${\mathcal{H}}^d[G\times SO\left(\infty \right),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed SPT phases in $d$-dimensional space-time are described by $E^d\left(G\right)⋊{\oplus }_{k=1}^{d-1}{\mathcal{H}}^k(G,{\text{iTO}}_L^{d-k})\oplus {\mathcal{H}}^d(G,\mathbb{R}/\mathbb{Z})$ where $G$ may contain time reversal. Here ${\text{iTO}}_L^d$ is the set of the topologically ordered phases in $d$-dimensional space-time that have no topological excitations, and one has ${\text{iTO}}_L^1={\text{iTO}}_L^2={\text{iTO}}_L^4={\text{iTO}}_L^6=0,{\text{iTO}}_L^3=\mathbb{Z},{\text{iTO}}_L^5={\mathbb{Z}}_2,{\text{iTO}}_L^7=2\mathbb{Z}$. For $G=U\left(1\right)⋊Z_2^T$ (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond ${\mathcal{H}}^d[U\left(1\right)⋊Z_2^T,\mathbb{R}/\mathbb{Z}]$ are described by ${\mathbb{Z}}_2$ in 3 + 1D, $\mathbb{Z}$ in 4 + 1D, $3{\mathbb{Z}}_2$ in 5 + 1D, and $4{\mathbb{Z}}_2$ in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

Figure 1

(a) A loop creation. (b) A loop annihilation. (c) A line reconnection.

Figure 2

(a) A point is split into three points. (b) A surface $N^2$ is split into three surfaces $N_1^2,N_2^2,N_3^2$.

Figure 3

Loop annihilation: (a) as we shrink the black circle to a point, the black line sweeps across the intersection of the red and blue lines once. This means that $N_1^2,N_2^2,N_3^2$ 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 $N_1^2,N_2^2,N_3^2$ intersect once in the line reconnection process.

Figure 4

(a) A $Z_2$-symmetry twist on a torus. (c) The $Z_2$-symmetry twist obtained from (a) by double Dehn twist. $\left(\to \to \mathrm{c}\right)$ contains a line reconnection.

Figure 5

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 one of the defects by ${360}^{\circ }$. The combination of those moves does not generate any phase since the number of the reconnection move is even.

Figure 6

Two identical $Z_2$ monodromy defects on $S^2$. The boundary across which we do the $Z_2$ twist is split into the red and blue curves. Note that the splitting is identical at the two monodromy defects. The red and blue lines cross once, indicating that ${\int }_{S^2}{a}_1^2=1$.

Figure 7

The shaded disk represents a two-dimensional manifold $\mathbb{R}{P}^2$, where the opposite points on the boundary are identified $\left(\widehat{\mathit{r}}\sim -\widehat{\mathit{r}}\right)$. The blue and red lines are two noncontractible loops in $\mathbb{R}{P}^2$. Consider a $Z_2$ twist $a_1$ described by ${\left[a_1\right]}^{*}=$ a contractible loop. Then the blue and red lines represent the same $Z_2$ twist $a_1$. For such a $Z_2$ twist, we find that ${\int }_{\mathbb{R}{P}^2}{a}_1^2=1$ since the blue and red lines cross once. The above $Z_2$ twist is also the orientation reversing twist. So $a_1={\text{w}}_1$ and we have ${\int }_{\mathbb{R}{P}^2}{\text{w}}_1^2=1$.

Figure 8

(a) The path integral of a single boundary point of 1D space over the time loop $S^1$. The shaded area represents the 1 + 1D space-time. The two ends of the thick line are identified to form a loop $S^1$. The two blue lines are also identified. (b) The loop $S^1$ is extended to a sphere with a hole. The identified blue line is also shown.

Figure 9

(a) The path integral of a single boundary point over the time loop $S^1$ with two time-reversal transformations at points 0 and 1. The shaded area represents the 1 + 1D space-time. The two ends of the thick line are identified to form a loop $S^1$. The two blue lines are also identified after a horizontal reflection. The two red lines on the two sides of the thick line are identified as well after a horizontal reflection. (b) The shaded disk represents a two-dimensional manifold $\mathbb{R}{P}^2$, where the opposite points on the boundary are identified $\left(\widehat{\mathit{r}}\sim -\widehat{\mathit{r}}\right)$. The loop $S^1$ in (a) is extended to the $\mathbb{R}{P}^2$ with a hole in (b). The two red lines and the two blue lines in (a) are also shown in (b).

Figure 10

Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.