Bulk Emergence and the RG Flow of Entanglement Entropy

TL;DR

This paper develops perturbative methods to calculate entanglement entropy in deformed conformal field theories, revealing universal terms, an emergent higher-dimensional bulk description, and a connection to gravitational dynamics.

Contribution

It introduces a universal second-order correction to entanglement entropy and links it to an emergent AdS bulk and gravitational equations, generalizing previous holographic insights.

Findings

Universal second-order entanglement entropy terms confirmed

Bulk modular Hamiltonian expressed as a local integral in emergent AdS space

Entanglement entropy related to area of a metric satisfying Einstein's equations

Abstract

We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional \emph{bulk} modular hamiltonian in an emergent $AdS$ space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein's equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.