Tensor networks with a twist: Anyon-permuting domain walls and defects in projected entangled pair states

We study the realization of anyon-permuting symmetries of topological phases on the lattice using tensor networks. Working on the virtual level of a projected entangled pair state, we find matrix product operators (MPOs) that realize all unitary topological symmetries for the toric and color codes. These operators act as domain walls that enact the symmetry transformation on anyons as they cross. By considering open boundary conditions for these domain wall MPOs, we show how to introduce symmetry twists and defect lines into the state.

Figure 1

Domain walls $S_P$ induce a permutation $P$ on the set of anyons $\left\{a_i\right\}$ when they cross. To ensure the topological phase is the same on both sides of the wall, only permutations that preserve the matrices $\mathcal{S}$ and $\mathcal{T}$ of Eq. (1) are allowed.

Figure 2

The toric code is defined on a square lattice, with spins on edges. It is convenient to write a PEPS tensor for each shaded plaquette. By inserting a bond at the vertex connecting adjacent gray plaquettes, a 4.8.8 lattice is obtained.

Figure 3

Applying a transversal (on-site) symmetry to regions $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$ (a) corresponds to domain walls at the boundaries $\partial \mathcal{R}$ and $\partial {\mathcal{R}}^{\prime }$ (b). If the symmetry is transversal, is the same as applying the symmetry to $\mathcal{R}\cup {\mathcal{R}}^{\prime }$ (c), so the domain walls should merge (d). For a nontransversal symmetry, a nontrivial operator will remain along the merge.

Figure 4

When merging two domain walls, the lattice geometry is altered along the wall. This is because the symmetry does not act transversally. Along the interface, a lattice translation is implemented. The colors correspond to those in the tensor network of Eq. (22).

Figure 5

To apply the APS to a region containing anyons, for example a pair of $e$ excitations (a), we need to deform the domain wall [using Eq. (24)] so that the interior contains only vacuum (b). Once the domain wall MPO has been inserted, the anyon operators can be pushed through to the interior (c).

Figure 6

The color code is defined on a three-colorable lattice. In this paper, we consider the 4.8.8 lattice for simplicity. Qubits are located on the vertices. By applying a unitary transformation to each green square of qubits, the color code can be decoupled into a pair of toric codes (located on the dashed lattice) and a pair of local qubits. The PEPS tensors end up centered on the plaquettes indicated by ‘$\times$.'