Relative entropy and the RG flow

TL;DR

This paper links relative entropy between vacuum states of different theories to entanglement entropy differences, providing new proofs of the c-theorem and RG flow properties in various dimensions.

Contribution

It offers an alternative proof of the c-theorem in 2D and shows the decrease of the area term coefficient in entanglement entropy along RG flows in higher dimensions.

Findings

Relative entropy equals entanglement entropy difference on null surfaces.

Positivity and monotonicity of relative entropy imply c-theorem and RG flow results.

Conditions for convergence of relative entropy depend on dimensions and operator dimensions.

Abstract

We consider the relative entropy between vacuum states of two different theories: a conformal field theory (CFT), and the CFT perturbed by a relevant operator. By restricting both states to the null Cauchy surface in the causal domain of a sphere, we make the relative entropy equal to the difference of entanglement entropies. As a result, this difference has the positivity and monotonicity properties of relative entropy. From this it follows a simple alternative proof of the c-theorem in d=2 space-time dimensions and, for d>2, the proof that the coefficient of the area term in the entanglement entropy decreases along the renormalization group (RG) flow between fixed points. We comment on the regimes of convergence of relative entropy, depending on the space-time dimensions and the conformal dimension $\Delta$ of the perturbation that triggers the RG flow.