Entanglement entropy for (3+1)-dimensional topological order with excitations

Abstract

Excitations in (3+1)-dimensional [(3+1)D] topologically ordered phases have very rich structures. (3+1)D topological phases support both pointlike and stringlike excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the following question: How do 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 order? We are mainly interested in (3+1)D topological order that can be realized in Dijkgraaf-Witten (DW) gauge theories, which are labeled by a finite group G and its group 4-cocycle ω∈H⁴[G;U(1)] up to group automorphisms. We find that each topological excitation contributes a universal constant ln d[subscript i] to the entanglement entropy, where d[subscript i] is the quantum dimension that depends on both the structure of the excitation and the data (G,ω). The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory (G,ω). In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles ω from the others.