Structure of the Entanglement Entropy of (3+1)D Gapped Phases of Matter

TL;DR

This paper investigates the structure of entanglement entropy in (3+1)D gapped phases, establishing a framework to define and compute topological entanglement entropy and illustrating how it varies with model modifications.

Contribution

It provides an explicit definition of topological entanglement entropy in (3+1)D and demonstrates how it can be computed and affected by model changes, extending previous understanding.

Findings

Entanglement entropy's constant part reduces to sphere and torus contributions.

Adding twist terms to the Lagrangian can alter the topological entanglement entropy.

Ground state degeneracy on a 3-torus relates to topological entanglement entropy across a 2-torus.

Abstract

We study the entanglement entropy of gapped phases of matter in three spatial dimensions. We focus in particular on size-independent contributions to the entropy across entanglement surfaces of arbitrary topologies. We show that for low energy fixed-point theories, the constant part of the entanglement entropy across any surface can be reduced to a linear combination of the entropies across a sphere and a torus. We first derive our results using strong sub-additivity inequalities along with assumptions about the entanglement entropy of fixed-point models, and identify the topological contribution by considering the renormalization group flow; in this way we give an explicit definition of topological entanglement entropy $S_{\mathrm{topo}}$ in (3+1)D, which sharpens previous results. We illustrate our results using several concrete examples and independent calculations, and show adding "twist" terms to the Lagrangian can change $S_{\mathrm{topo}}$ in (3+1)D. For the generalized Walker-Wang models, we find that the ground state degeneracy on a 3-torus is given by $\exp(-3S_{\mathrm{topo}}[T^2])$ in terms of the topological entanglement entropy across a 2-torus. We conjecture that a similar relationship holds for Abelian theories in $(d+1)$ dimensional spacetime, with the ground state degeneracy on the $d$-torus given by $\exp(-dS_{\mathrm{topo}}[T^{d-1}])$.