Entanglement entropy of (3+1)D topological orders with excitations

TL;DR

This paper investigates how various topological excitations in (3+1)D topological orders influence entanglement entropy, revealing that excitations contribute universal constants related to their quantum dimensions and that entanglement entropy can distinguish different underlying topological data.

Contribution

It provides a detailed analysis of how excitations contribute to entanglement entropy in (3+1)D Dijkgraaf-Witten theories, linking entropy contributions to quantum dimensions and topological data.

Findings

Excitations contribute a universal constant ln d_i to entanglement entropy.

Linked and unlinked excitations encode different topological information.

Hopf-link loop excitations can distinguish certain 4-cocycles in the theory.

Abstract

Excitations in (3+1)D topologically ordered phases have very rich structures. (3+1)D topological phases support both point-like and string-like excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the question how different types of topological excitations contribute to the entanglement entropy, or alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological orders? We are mainly interested in (3+1)D topological orders that can be realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite group $G$ and its group 4-cocycle $\omega\in\mathcal{H}^4[G;U(1)]$ up to group automorphisms. We find that each topological excitation contributes a universal constant $\ln d_i$ to the entanglement entropy, where $d_i$ is the quantum dimension that depends on both the structure of the excitation and the data $(G,\,\omega)$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\,\omega)$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\omega$ from the others.