Exactly solvable models for symmetry-enriched topological phases

We construct fixed-point wave functions and exactly solvable commuting-projector Hamiltonians for a large class of bosonic symmetry-enriched topological (SET) phases, based on the concept of equivalent classes of symmetric local unitary transformations. We argue that for onsite unitary symmetries, our construction realizes all SETs free of anomaly, as long as the underlying topological order itself can be realized with a commuting-projector Hamiltonian. We further extend the construction to antiunitary symmetries (e.g., time-reversal symmetry), mirror-reflection symmetries, and to anomalous SETs on the surface of three-dimensional symmetry-protected topological phases. Mathematically, our construction naturally leads to a generalization of group extensions of unitary fusion categories to antiunitary symmetries.

Figure 1

A trivalent-graph lattice. The arrows on the links show the branching structure. Each plaquette is decorated by a group element ${\mathbf{g}}_i\in G$.

Figure 2

A triangulation of a 3D bulk, with a 2D surface. The bulk is represented by only one vertex, carrying a group element ${\mathbf{g}}_{*}$. The vertices carrying ${\mathbf{g}}_1,\cdots {\mathbf{g}}_4$ belong to the 2D surface.

Figure 3

Steps of deforming the right-hand side of Eq. (42) to its left-hand side. (a) Step 1: the initial configuration containing an inner loop carrying a topological charge $s$. The inner loop consists of counterpropagating segments carrying charges $s$ and $\overline{s}$, respectively. The dashed lines are vacuum strings carrying charge 0. The blue color indicates the locations where the $F$ moves will be applied to obtain the next configuration. (b) Step 2: the second configuration is obtained after applying six $F$ moves and $H$ moves (see Appendix pp2), located at the links marked by the blue color. The letter denotes whether an $F$ move or an $H$ move is performed. The red links mark the locations of the $F$ moves leading to the next configuration. (c) Step 3: the third configuration is obtained through six $F$ moves and $H$ moves located at the red links. The letter denotes whether an $F$ move or an $H$ move is performed. Finally, this configuration is changed into the one on the left-hand side of Eq. (42), by eliminating the bubbles using the move in Eq. (4).

Figure 4

Illustration of an open string operator $W^{\downarrow }$ and closed string operator $W^{\uparrow }$.