Symmetry enrichment in three-dimensional topological phases

While two-dimensional symmetry-enriched topological phases $\left(\text{SET}\mathrm{s}\right)$ have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry $G_s$ on gauge theories (denoted by $\text{GT})$ with gauge group $G_g$. The resulting symmetric gauge theories are dubbed “symmetry-enriched gauge theories” $\left(\text{SEG}\right)$, which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on $\text{SEG}\mathrm{s}$ with gauge group $G_g={\mathbb{Z}}_{N_1}\times {\mathbb{Z}}_{N_2}\times \cdots$ and onsite unitary symmetry group $G_s={\mathbb{Z}}_{K_1}\times {\mathbb{Z}}_{K_2}\times \cdots$ or $G_s=\mathrm{U}\left(1\right)\times {\mathbb{Z}}_{K_1}\times \cdots$. Each $\text{SEG}(G_g,G_s)$ is described in the path-integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground-state properties (i.e., $\text{SET}$ orders) of $\text{SEG}\mathrm{s}$ in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the mixed multiloop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from $\mathsf{\text{SEG}}\mathrm{s}$ to $\mathsf{\text{SET}}\mathrm{s}$. By giving full dynamics to background gauge fields, $\mathsf{\text{SEG}}\mathrm{s}$ may be eventually promoted to a set of new gauge theories (denoted by ${\mathsf{\text{GT}}}^{*})$. Based on their gauge groups, ${\mathsf{\text{GT}}}^{*}\mathrm{s}$ may be further regrouped into different classes, each of which is labeled by a gauge group $G_g^{*}$. Finally, a web of gauge theories involving $\mathsf{\text{GT}},\mathsf{\text{SEG}},\mathsf{\text{SET,}}$ and ${\mathsf{\text{GT}}}^{*}$ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.

Figure 1

Schematic representation of a web of gauge theories with global symmetry. We start with a discrete gauge group $G_g$ that generates several topologically distinct gauge theories $\left(\mathsf{\text{GT}}\mathrm{s}\right)$, one of which is untwisted. We then assign symmetry charge of symmetry group $G_s$ to topological currents of gauge theories through coupling to background gauge fields. There are usually many different ways of symmetry assignment, each of which is represented by a colored arrow. Within each specific symmetry assignment, we obtain many $\mathsf{\text{SEG}}\mathrm{s}$. For example, ${\mathsf{\text{SEG}}}_1,{\mathsf{\text{SEG}}}_2$, and ${\mathsf{\text{SEG}}}_3$ belong to the same symmetry assignment (marked by magenta arrows) in ${\mathsf{\text{GT}}}_1$. It is generically possible that some of symmetry assignments do not provide $\mathsf{\text{SEG}}$ descendants for $\mathsf{\text{GT}}$, which we mark as “N/A.” To identify $\mathsf{\text{SET}}\mathrm{s}$, we need to further study ground-state properties of $\mathsf{\text{SEG}}\mathrm{s}$. We may externally insert symmetry fluxes into the system and perform Aharonov-Bohm experiments and the mixed version of three-loop braiding experiments. Two $\mathsf{\text{SEG}}\mathrm{s}$ (e.g., ${\mathsf{\text{SEG}}}_3$ and ${\mathsf{\text{SEG}}}_4)$ may possibly describe the same $\mathsf{\text{SET}}$ phase. Further, by giving full dynamics to the background gauge fields, the resulting new gauge theories (denoted by ${\mathsf{\text{GT}}}^{*})$ are generated. Since basis transformations are allowed, there should be in general many-to-one correspondence between $\mathsf{\text{SEG}}\mathrm{s}$ and ${\mathsf{\text{GT}}}^{*}\mathrm{s}$. Finally, all ${\mathsf{\text{GT}}}^{*}\mathrm{s}$ may be regrouped via identifying their gauge groups (denoted by $G_g^{*})$.

Figure 2

A schematic representation of “layers” (Sec. 2a) and the general procedure (Sec. 2c). Each “layer” denotes a 3D system. It should be noted that all layers are stacked together in the same 3D spatial region although they are not so in this figure. $\mathsf{\text{GT}}$ before imposing symmetry resides in type-I layers. Type-II layers are described by level-1 BF terms before imposing symmetry. By “trivial,” we mean that these layers do not carry gauge groups. The dashed curves represent topological interactions between layers. Actually, three- and four-layer topological interactions should also be considered.

Figure 3

A mixed version of three-loop braiding process among gauge fluxes and symmetry fluxes in $\mathsf{\text{SEG}}({\mathbb{Z}}_2,{\mathbb{Z}}_2)$ of Table 2. Loops in red and black denote symmetry flux loop $\left(\sigma \right)$ and gauge flux loop $\left({\mathrm{\Sigma }}^1\right)$, respectively. The dashed curves shows the trajectory of one loop that moves around another loop, both of which are linked to the base loop.

Figure 4

Two concrete examples of webs of gauge theories shown in Fig. 1. $\mathsf{\text{SEG}}({\mathbb{Z}}_2,{\mathbb{Z}}_2)$ and $\mathsf{\text{SEG}}({\mathbb{Z}}_2,{\mathbb{Z}}_3)$ are shown in (a) and (b), respectively. ${\mathsf{\text{SEG}}}_1$ in (a) can be found in Table 1. ${\mathsf{\text{SEG}}}_{2,\cdots ,5}$ in (a) can be found in Table 2. ${\mathsf{\text{SEG}}}_1$ in (b) can be found in Table 1 by setting $K=3$. ${\mathsf{\text{SEG}}}_2$ in (b) can be found in the first subtable of Table 2 by setting $K=3$.

Figure 5

A skeleton of the web of gauge theories for $\mathsf{\text{SEG}}({\mathbb{Z}}_2\times {\mathbb{Z}}_2,{\mathbb{Z}}_2)$.