Topological entanglement entropy meets holographic entropy inequalities

TL;DR

This paper analyzes topological entanglement entropy (TEE), explains the effectiveness of subtraction schemes for its computation, generalizes these schemes for multiple regions, and shows holographic entropy inequalities hold for topologically ordered systems.

Contribution

It clarifies why certain information quantities detect TEE, generalizes subtraction schemes for arbitrary regions, and connects holographic entropy inequalities with topological order.

Findings

Subtraction schemes work for TEE computation and are generalized for multiple regions.

Conditions for information quantities to capture TEE are explicitly identified.

Holographic entropy inequalities are satisfied by ground states of topologically ordered systems.

Abstract

Topological entanglement entropy (TEE) is an efficient way to detect topological order in the ground state of gapped Hamiltonians. The seminal work of Kitaev and Preskill~\cite{preskill-kitaev-tee} and simultaneously by Levin and Wen~\cite{levin-wen-tee} proposed information quantities that can probe the TEE. In the present work, we explain why the subtraction schemes in the proposed information quantities~\cite{levin-wen-tee,preskill-kitaev-tee} work for the computation of TEE and generalize them for arbitrary number of subregions by explicitly noting the necessary conditions for an information quantity to capture TEE. Our conditions differentiate the probes defined by Kitaev-Preskill and Levin-Wen into separate classes. While there are infinitely many possible probes of TEE, we focus particularly on the cyclic quantities $Q_{2n+1}$ and multi-information $I_n$. We also show that the holographic entropy inequalities are satisfied by the quantum entanglement entropy of the non-degenerate ground state of a topologically ordered two-dimensional medium with a mass gap.