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

TL;DR

This work demonstrates that the topological entanglement entropy for a 2+1D topological order with an entanglement cut touching gapped boundaries is governed by half-linking numbers $\gamma_{xc}$ through twisted characters, generalizing the role of the $S$-matrix in boundary-free settings. The authors develop both open-string Ishibashi and closed-string boundary-state formalisms, related by Cardy-like overlaps, and show that the resulting entropy features a universal topological term $\ln\gamma_{x0}$ (or related expressions) plus nonuniversal area terms. In Abelian Chern-Simons theories, explicit calculations reveal how boundary condensates (electric/magnetic) and their junctions contribute via twisted-character normalization, with Majorana zero modes producing characteristic $-\ln\sqrt{2}$ terms in the e–m case. A condensed-confined duality for 2×2 K-matrix theories is established, clarifying how shared condensed sectors determine the open/closed-channel data, and highlighting conditions under which $\gamma$ remains unitary. The results offer a coherent framework to understand boundary effects on entanglement, suggest generalizations to more complex gapped boundaries, and hint at holographic analogies with bulk–boundary correspondences.

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.

Figures (5)

• Figure 1: Ground state basis states on a cylinder.
• Figure 2: "Open string" modes in the folded picture and "closed string" modes in the unfolded picture.
• Figure 3: $K\Omega_{\mu\nu}$ and $\Omega_{\mu\nu}$ are invariant subspaces of $AA^T$
• Figure 4: 1D sublattices of the charge lattice $\mathbb{Z}^2$
• Figure 5: The $4$ boundary condensates of $D(S_3)$ Dijkgraaf-Witten model, related by the $C\leftrightarrow F$ duality and the condensed-confined duality.