Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy II -- Cutting through the boundary

TL;DR

This paper analyzes entanglement entropy in 2+1D topological orders with gapped boundaries, revealing how boundary effects modify entropy calculations and introducing boundary states based on half-linking numbers.

Contribution

It introduces a novel approach to compute entanglement entropy with boundary cuts using half-linking numbers and constructs boundary states beyond diagonal RCFTs.

Findings

Entanglement entropy is influenced by half-linking numbers at boundaries.

Boundary states can be constructed from half-linking numbers, providing a new perspective.

The approach applies to Abelian Chern-Simons theories with specific boundary conditions.

Abstract

We compute the entanglement entropy in a 2+1 dimensional topological order in the presence of gapped boundaries. Specifically, we consider entanglement cuts that cut through the boundaries. We argue that based on general considerations of the bulk-boundary correspondence, the "twisted characters" feature in the Renyi entropy, and the topological entanglement entropy is controlled by a "half-linking number" in direct analogy to the role played by the S-modular matrix in the absence of boundaries. We also construct a class of boundary states based on the half-linking numbers that provides a "closed-string" picture complementing an "open-string" computation of the entanglement entropy. These boundary states do not correspond to diagonal RCFT's in general. These are illustrated in specific Abelian Chern-Simons theories with appropriate boundary conditions.