Entanglement entropy from one-point functions in holographic states

TL;DR

This paper develops a perturbative framework to compute entanglement entropy in holographic CFT states near the vacuum, incorporating second-order corrections from one-point functions and explicitly deriving formulas for two-dimensional cases.

Contribution

It introduces a second-order expansion formula for entanglement entropy in holographic states, extending previous first-order results and connecting CFT calculations with bulk gravitational energy.

Findings

Derived a second-order entanglement entropy formula for arbitrary regions.

Explicitly computed the second-order contribution from the stress tensor in 2D CFTs.

Reproduced the stress tensor formula via conformal transformations and scalar wave equations.

Abstract

For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk fields. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the first-order result given by the entanglement first law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the first-order result in arXiv/1509.00113.