Models for gapped boundaries and domain walls

TL;DR

The work builds a unified dictionary linking Levin-Wen lattice models for 2D topological phases with gapped boundaries and domain walls to tensor-categorical data, showing that bulk excitations live in $Z(C)$ while boundaries and walls are described by module and bimodule structures, respectively. Invertible (transparent) walls induce braided equivalences between bulk centers $Z(C)$ and $Z(D)$, and excitations at all codimensions are modeled by module functors and natural transformations, with a concrete toric code example illustrating the construction. The authors also connect this framework to extended Turaev-Viro topological field theory and discuss higher-dimensional generalizations via higher categories and dualizable objects. Overall, the paper provides a detailed physical-to-categorical dictionary and suggests a path toward a fully extended, higher-dimensional theory of topological phases and their defects.

Abstract

We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category $\calC$ as in the Levin-Wen model, whereas the boundary is associated with a module category over $\calC$. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

Figures (8)

• Figure 1: Toric code with boundaries of two types and a defect line.
• Figure 2: An oriented planar graph with edge and vertex labels.
• Figure 3: The action of the plaquette operator $B_{\mathbf{p}}^k$: a) the initial state of the plaquette; b) a symbolic representation of the operator $B_{\mathbf{p}}^k$ applied to it; c) the loop is partially fused using Eq. (\ref{['eq:decomp_id']}) (some labels and the overall factor are not shown); d) the corner triangles have been evaluated to trivalent vertices (summation over $j_p'$, $\alpha_q'$ is assumed).
• Figure 4: A neighborhood of the boundary in a Levin-Wen model.
• Figure 5: Boundary excitations (the unexcited part of the lattice is shown in gray): a) the labels characterizing an excited state; b) operator $B_{\mathbf{p}}$ acts on the adjacent plaquette; c) the loop is partially fused; d) the loop is completely fused.