Entanglement Renyi entropies in holographic theories

TL;DR

This paper investigates entanglement Renyi entropies in holographic theories, correcting previous calculations, and provides evidence supporting phase transition predictions and the equivalence of certain theories.

Contribution

It corrects earlier errors in computing EREs holographically and extends the analysis to two-interval cases, supporting phase transition and mutual information predictions.

Findings

Gravity computations match known CFT results for single intervals.

Evidence supports phase transition in entanglement entropy for two intervals.

Large-N symmetric-product theories share EREs with holographic theories.

Abstract

Ryu and Takayanagi conjectured a formula for the entanglement (von Neumann) entropy of an arbitrary spatial region in an arbitrary holographic field theory. The von Neumann entropy is a special case of a more general class of entropies called Renyi entropies. Using Euclidean gravity, Fursaev computed the entanglement Renyi entropies (EREs) of an arbitrary spatial region in an arbitrary holographic field theory, and thereby derived the RT formula. We point out, however, that his EREs are incorrect, since his putative saddle points do not in fact solve the Einstein equation. We remedy this situation in the case of two-dimensional CFTs, considering regions consisting of one or two intervals. For a single interval, the EREs are known for a general CFT; we reproduce them using gravity. For two intervals, the RT formula predicts a phase transition in the entanglement entropy as a function of their separation, and that the mutual information between the intervals vanishes for separations larger than the phase transition point. By computing EREs using gravity and CFT techniques, we find evidence supporting both predictions. We also find evidence that large-$N$ symmetric-product theories have the same EREs as holographic ones.