Entanglement in BF theory I: Essential topological entanglement

TL;DR

This paper introduces a refined concept of topological entanglement entropy in Abelian BF theories, capturing more intricate topological features and providing a finite, intrinsic measure of entanglement in higher-dimensional topological orders.

Contribution

It defines and analyzes the notion of essential topological entanglement in Abelian BF theories, extending the understanding of topological entanglement beyond traditional measures in higher dimensions.

Findings

Essential topological entanglement is finite and positive.

It is sensitive to detailed topological features of the state.

The framework applies to arbitrary dimensions in Abelian topological orders.

Abstract

We study the entanglement structure of Abelian topological order described by $p$-form BF theory in arbitrary dimensions. We do so directly in the low-energy topological quantum field theory by considering the algebra of topological surface operators. We define two appropriate notions of subregion operator algebras which are related by a form of electric-magnetic duality. To each subregion algebra we assign an entanglement entropy which we coin essential topological entanglement. This is a refinement to the traditional topological entanglement entropy. It is intrinsic to the theory, inherently finite, positive, and sensitive to more intricate topological features of the state and the entangling region. This paper is the first in a series of papers investigating entanglement and topological order in higher dimensions.