Topological invariants for gauge theories and symmetry-protected topological phases

We study the braiding statistics of particlelike and looplike excitations in two- (2D) and three-dimensional (3D) gauge theories with finite, Abelian gauge group. The gauge theories that we consider are obtained by gauging the symmetry of gapped, short-range entangled, lattice boson models. We define a set of quantities, called topological invariants, that summarize some of the most important parts of the braiding statistics data for these systems. Conveniently, these invariants are always Abelian phases, even if the gauge theory supports excitations with non-Abelian statistics. We compute these invariants for gauge theories obtained from the exactly soluble group cohomology models of Chen, Gu, Liu, and Wen, and we derive two results. First, we find that the invariants take different values for every group cohomology model with finite, Abelian symmetry group. Second, we find that these models exhaust all possible values for the invariants in the 2D case, and we give some evidence for this in the 3D case. The first result implies that every one of these models belongs to a distinct symmetry-protected topological (SPT) phase, while the second result suggests that these models may realize all SPT phases. These results support the group cohomology classification conjecture for SPT phases in the case where the symmetry group is finite, Abelian, and unitary.

Figure 1

Space-time trajectories of the vortices in the braiding processes associated with ${\Theta }_{ij}$ [(a); $N^{ij}=3$] and ${\Theta }_{ijk}$ [(b)]. The arrow of time is upward.

Figure 2

Three-loop braiding process. (a) The gray curves show the paths of two points on the moving loop $\alpha$. (b) Cross section of the braiding process in the plane that $\gamma$ lies in. (c) A torus ${\Omega }_{\alpha }$ is swept out by $\alpha$ during the braiding, which encloses the loop $\beta$ (dashed circle).

Figure 3

Fusion of two loops ${\beta }_1$ and ${\beta }_2$, both linked to $\gamma$. We denote this type of fusion by ${\beta }_1\times {\beta }_2$. (This is different notation from Ref. [24], where this type of fusion was denoted by ${\beta }_1+{\beta }_2$.)

Figure 4

(a) A thickened torus ${\mathbb{T}}^2\times [0,1]$ with a flux $\varphi$ threading the inner hole. (b) The thickened torus drawn as a cube with the top and bottom faces as well as the front and back faces identified.

Figure 5

A Borromean ring obtained by closing up the trajectories in Fig. 1.

Figure 6

Thought experiment to prove the constraint (55c) ($N_k=3$ is taken for illustration). For clarity, we split the composite braiding into two steps (i) and (ii). In the thought experiment, (ii) follows immediately after (i).

Figure 7

Fusion of two loops ${\beta }_1$ and ${\beta }_2$ that carry the same amount of flux ${\varphi }_{{\beta }_1}={\varphi }_{{\beta }_2}$ and that are linked to different base loops. We denote this type of fusion by ${\beta }_1\circ {\beta }_2$. This is different notation from Ref. [24], where this type of fusion was denoted by ${\beta }_1\oplus {\beta }_2$.

Figure 8

Diagrammatic proof that the unitary matrix associated with the braiding process in the definition of ${\Theta }_{ijk}$ is an Abelian phase.