Entanglement entropy of excited states in conformal perturbation theory and the Einstein equation

TL;DR

This paper studies how entanglement entropy in excited states of conformal field theories behaves under perturbations, revealing new scaling terms that challenge existing conjectures related to the Einstein equation and vacuum entropy.

Contribution

It provides a perturbative analysis of excited state entanglement entropy in deformed CFTs, identifying dominant scaling terms that suggest modifications to Jacobson's conjecture.

Findings

Entanglement entropy includes terms scaling as R^{2Δ}

For Δ ≤ d/2, these terms dominate the entropy behavior

Results imply a need to revise the conjecture relating entropy and Einstein equations

Abstract

For a conformal field theory (CFT) deformed by a relevant operator, the entanglement entropy of a ball-shaped region may be computed as a perturbative expansion in the coupling. A similar perturbative expansion exists for excited states near the vacuum. Using these expansions, this work investigates the behavior of excited state entanglement entropies of small, ball-shaped regions. The motivation for these calculations is Jacobson's recent work on the equivalence of the Einstein equation and the hypothesis of maximal vacuum entropy [arXiv:1505.04753], which relies on a conjecture stating that the behavior of these entropies is sufficiently similar to a CFT. In addition to the expected type of terms which scale with the ball radius as $R^d$, the entanglement entropy calculation gives rise to terms scaling as $R^{2\Delta}$, where $\Delta$ is the dimension of the deforming operator. When $\Delta\leq\frac{d}{2}$, the latter terms dominate the former, and suggest that a modification to the conjecture is needed.