Relative entropy and the RG flow

TL;DR

The paper develops a framework to compare vacuum states of a UV CFT and its RG-flow perturbed version using relative entropy reduced to a null Cauchy surface on a sphere. By mapping algebras across Cauchy surfaces and employing a conformal interaction picture along with a sphere modular Hamiltonian, the authors show that the relative entropy reduces to the entanglement-entropy difference on the null surface within a Δ-window, yielding a clean information-theoretic interpretation of RG irreversibility. This leads to a concise proof of Zamolodchikov’s c-theorem in d=2 and a monotonic decrease of the entanglement-entropy area term in d>2 along RG flows (within Δ < (d+2)/2). The results unify RG monotonicity with quantum information principles and offer potential insights into gravity and holography through the area-term behavior of entanglement entropy.

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 $Δ$ of the perturbation that triggers the RG flow.

Figures (2)

• Figure 1: Different Cauchy surfaces $\Sigma$ and $\Sigma'$ with the same causal domain of dependence $\mathcal{D}$; $\Sigma_{gl}$ is a global Cauchy surface.
• Figure 2: Modular flow vector field $\xi^\mu$ in the $d=2$ causal diamond of a spatial sphere. $\Delta \langle{\cal H}\rangle_{\Sigma}$ is described by the flux of $\xi^\mu$ through the Cauchy surface $\Sigma$. $\Delta \langle{\cal H}\rangle_{\Sigma_{\text{null}}} = 0$ on the null Cauchy surface $\Sigma_{\text{null}}$ for the range of perturbations discussed in the text. The divergence of $\xi^\mu$ integrated over the shaded region gives $\Delta {\cal H}_{\Sigma^\prime}-\Delta {\cal H}_\Sigma$ and hence the variation of the relative entropy with the surface.