Entanglement Temperature and Perturbed AdS$_3$ Geometry

TL;DR

This paper explores how entanglement entropy and temperature in a conformal field theory are affected by external perturbations, establishing a relationship with bulk AdS geometry via the AdS/CFT correspondence.

Contribution

It introduces a novel framework linking local entanglement temperature changes to bulk metric perturbations in AdS$_3$ using numerical methods.

Findings

Entanglement temperature varies with external perturbations.

Bulk metric perturbations can be reconstructed from entanglement data.

Numerical methods successfully relate boundary perturbations to bulk geometry.

Abstract

In analogy to the first law of thermodynamics, the increase in entanglement entropy $\delta S$ of a conformal field theory (CFT) is proportional to the increase in energy, $\delta E$, of the subsystem divided by an effective entanglement temperature, $T_E$. Extending this analogy, we study entanglement entropy when the subsystem is perturbed by applying an external field, expressed as a coupling to a local marginal operator in the CFT. We show that the resulting entropy change is associated with a change in the entanglement temperature itself, leading to an equation analogous to the Clausius relation. Using AdS/CFT duality we develop a relationship between a perturbation in the local entanglement temperature, $\delta T_E(x)$ of the CFT and the perturbation of the bulk AdS metric. Using the AdS$_3$ minimal surface as a probe, we can construct bulk metric perturbations from an exact numerical computation of the entanglement temperature in a two dimensional $c=1$ boundary theory deformed by a marginal perturbation.