Topological Entanglement Entropy and Holography

TL;DR

This paper investigates the holographic entanglement entropy in confining theories, revealing that topological entanglement entropy vanishes in 2+1 dimensions but is nonzero and cutoff-dependent in 3+1 dimensions, providing insights into topological order.

Contribution

It introduces a holographic approach to compute topological entanglement entropy in confining theories, highlighting differences between 2+1 and 3+1 dimensions.

Findings

Topological entanglement entropy vanishes in 2+1D confining gauge theories.

In 3+1D, the topological entanglement entropy is nonzero and depends on the cutoff.

Two types of bulk hypersurfaces dominate entanglement entropy calculations depending on the radius.

Abstract

We study the entanglement entropy in confining theories with gravity duals using the holographic prescription of Ryu and Takayanagi. The entanglement entropy between a region and its complement is proportional to the minimal area of a bulk hypersurface ending on their border. We consider a disk in 2+1 dimensions and a ball in 3+1 dimensions and find in both cases two types of bulk hypersurfaces with different topology, similar to the case of the slab geometry considered by Klebanov, Kutasov and Murugan. Depending on the value of the radius, one or the other type of hypersurfaces dominates the calculation of entanglement entropy. In 2+1 dimensions a useful measure of topological order of the ground state is the topological entanglement entropy, which is defined to be the constant term in the entanglement entropy of a disk in the limit of large radius. We compute this quantity and find that it vanishes for confining gauge theory, in accord with our expectations. In 3+1 dimensions the analogous quantity is shown to be generically nonzero and cutoff-dependent.