Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions

TL;DR

This paper analyzes entanglement entropy in gapped phases across various dimensions, revealing how topological order influences entropy contributions and identifying new types of topological entanglement entropy in higher dimensions.

Contribution

It introduces a curvature-based expansion for entanglement entropy in trivial phases and characterizes the types of topological entanglement entropy in three and higher dimensions.

Findings

In trivial phases, entanglement entropy can be expanded via boundary curvature.

In 3D, TEE depends linearly on the number of boundary components.

Higher-dimensional TEE depends on higher Betti numbers.

Abstract

We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an `entropy density' over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size, and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size independent constant contribution to the entanglement entropy can appear for trivial phases in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE) that cannot be obtained by adding local contributions. We find that in three dimensions there is just one type of TEE, as in two dimensions, that depends linearly on the number of connected components of the boundary (the `zeroth Betti number'). In D > 3 dimensions, new types of TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in D >=3.