Edge theory approach to topological entanglement entropy, mutual information and entanglement negativity in Chern-Simons theories

TL;DR

This paper introduces an edge theory approach to compute entanglement entropy, mutual information, and negativity in (2+1)D topologically ordered phases, revealing topological data and distinguishing Abelian from non-Abelian orders.

Contribution

It presents a new edge theory framework applicable to arbitrary genus manifolds, enabling analysis of fusion, braiding, and interference effects in topological phases.

Findings

Edge theory method computes entanglement measures for topological phases.

Entanglement negativity distinguishes Abelian from non-Abelian orders.

Method detects topological data like R-symbols and monodromy.

Abstract

We develop an approach based on edge theories to calculate the entanglement entropy and related quantities in (2+1)-dimensional topologically ordered phases. Our approach is complementary to, e.g., the existing methods using replica trick and Witten's method of surgery, and applies to a generic spatial manifold of genus $g$, which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the $R$-symbols, monodromy and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent non-contractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.