Entanglement Temperature and Perturbed AdS3 Geometry
Abstract
In analogy to the first law of thermodynamics, the increase in entanglement entropy δ S of a conformal field theory (CFT) is proportional to the increase in energy, δ E, of the subsystem divided by an effective entanglement temperature, TE. 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, δ TE(x) of the CFT and the perturbation of the bulk AdS metric. Using the AdS3 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.