Correspondence between bulk entanglement and boundary excitation spectra in 2d gapped topological phases

TL;DR

This paper explores the relationship between boundary excitation spectra and bulk entanglement spectra in 2D gapped topological phases, revealing how boundary conditions influence entanglement properties in string-net models.

Contribution

It establishes a detailed correspondence between boundary spectra and bulk entanglement spectra in non-chiral topological orders, including explicit examples with toric code and Fibonacci models.

Findings

Boundary spectra correspond to entanglement spectra under specific conditions.

Different boundary conditions lead to distinct entanglement cut interpretations.

Explicit examples demonstrate the correspondence in various topological models.

Abstract

We study the correspondence between boundary spectrum of non-chiral topological orders on an open manifold $\mathcal{M}$ with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed manifold. The closed manifold is bipartitioned into two subsystems, one of which has the same topology as $\mathcal{M}$. Specifically, we focus on the case of generalized string-net models and discuss the cases where $\mathcal{M}$ is a disk or a cylinder. When $\mathcal{M}$ has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code, $\mathbb{Z}_N$ theories, and the simple non-abelian case of doubled Fibonacci are demonstrated.