# Topological Entanglement Entropy in \(d\)-dimensions for Abelian Higher Gauge Theories

## TL;DR

This paper derives a general formula for topological entanglement entropy in higher-dimensional Abelian gauge theories, revealing how topology influences entanglement and extending understanding of topological phases.

## Contribution

It introduces a universal method to compute entanglement entropy in higher-dimensional Abelian higher gauge theories, linking it to ground state degeneracy and topology.

## Key findings

- Entanglement entropy proportional to log of ground state degeneracy.
- Topological contributions depend on the topology of the entangling surface.
- Formalism applicable to arbitrary dimensions, consistent with known models.

## Abstract

We compute the topological entanglement entropy for a large set of lattice models in $d$-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of $d$-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy $S_A$ in a sub-region $A$ is proportional to $\log(GSD_{\tilde{A}})$, where \(GSD_{\tilde{A}}\) is the ground state degeneracy of a particular restriction of the full model to \(A\). The quantity $GSD_{\tilde{A}}$ can be further divided into a contribution that scales with the size of the boundary $\partial A$ and a term which depends on the topology of $\partial A$. There is also a topological contribution coming from $A$ itself, that may be non-zero when $A$ has a non-trivial homology. We present some examples and discuss how the topology of $A$ affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension $d$. The result is in agreement with entanglement calculations for known topological models.