Classification and properties of symmetry-enriched topological phases: Chern-Simons approach with applications to $Z_2$ spin liquids

We study (2+1)-dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry-enriched topological (SET) phases. Here, we present a $K$-matrix Chern-Simons approach to identify distinct phases with Abelian topological order, in the presence of unitary or antiunitary global symmetries. A key step is the identification of a smooth edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method by classifying $Z_2$ topological order ($Z_2$ spin liquids) in the presence of an internal $Z_2$ global symmetry for which we find six distinct phases. These include two phases with an unconventional action of symmetry that permutes anyons leading to symmetry-protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry-enriched Laughlin states and double-semion theories are also discussed. Somewhat surprisingly, we observe that (i) gauging the global symmetry of distinct SET phases leads to topological orders with the same total quantum dimension, and (ii) a pair of distinct SET phases can yield the same topological order on gauging the symmetry.

Figure 1

Edge sewing criterion to distinguish symmetry-enriched topological (SET) phases. Only the microscopic degrees of freedom, i.e., “electrons” (and not gauge charged objects such as anyons/fractionalized quasiparticles), can tunnel between the two edges of a pair of semi-infinite cylinders. If two SET phases can be continuously tuned into one another without a phase transition (while preserving symmetry), there is a “smooth” sewing between the two cylinders of SET phases No. 1 and No. 2. This implies that all edge excitations are gapped by a few symmetry-allowed terms that tunnel “electrons” between the two edges. In the thermodynamic limit these tunneling terms lead to $M$ degenerate ground states, corresponding exactly to the $M$-fold torus degeneracy of the topological order. On the other hand, if the two SET phases are different, there is no such “smooth” boundary condition to sew the two edges. A precise version of this statement is formulated in Criterion I in Sec. 2e.

Figure 2

A fermion mode ($f$) localized at the boundary between two subsystems $A$ and $B$ which from a bipartition of the on a sphere, where the unconventional Ising symmetry-enriched $Z_2$ spin liquid resides. Under the Ising symmetry operation, an electric charge $e$ will transform into a magnetic vortex $m$. Consider one electric charge is created in each subsystem. If we perform Ising ($Z_2$) symmetry only on subsystem $A$, a fermion mode will emerge on the boundary, as the electric $e$ charge turns into a magnetic vortex $m$ in $A$.

Figure 3

The Ising symmetry eigenstates are linear combinations of minimal entropy states (MESs) for an unconventional Ising symmetry-enriched $Z_2$ spin liquid since one MES $|e\rangle$ transforms into another MES $|m\rangle$ under Ising symmetry operation.

Figure 4

Domain-wall bound state on the edge of unconventional Ising symmetry-enriched $Z_2$ spin liquids (see Table 3). In these SET phases, under $Z_2$ symmetry operation, one electric charge will transform into a magnetic vortex and vice versa. The onsite unitary $Z_2$ (Ising) symmetry can be, e.g., a spin-flip symmetry. On the two sides of the Ising symmetry domain wall, two different backscattering “mass” terms related by spin-flip Ising symmetry are added to gap out the edge states. These two mass terms break $Z_2$ symmetry in opposite ways. A non-Abelian bound state with quantum dimension $d_{q_{\mathit{g}}}=\sqrt{2}$ is localized at each Ising domain wall. For a conventional $Z_2$ SET phase, such an Ising mass domain wall will trap an Abelian bound state with quantum dimension 1.

Figure 5

The process in which a symmetry flux $q_{\mathit{g}}$ and its antiparticle ${\overline{q}}_{\mathit{g}}$ are created out of the vacuum, dragged around a quasiparticle $a$, and then annihilated. The dashed line denotes the trajectory of $q_{\mathit{g}}$ and ${\overline{q}}_{\mathit{g}}$ from their creation to annihilation. After this process, symmetry $\mathit{g}$ is implemented in the region inside the closed loop (dashed line), and hence quasiparticle $a$ is transformed into $b$ under $\mathit{g}$ symmetry operation.