Entanglement Temperature and Perturbed AdS${}_3$ 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.

5 pages, 3 figures; added reference; expanded conclusion; as published in PRD

References in corpus (2)

• Towards a derivation of holographic entanglement entropy
• Gravity Dual of Quantum Information Metric

Cited by in corpus (1)

• Generalized geodesic deviation equations and an entanglement first law for rotating BTZ black holes