Comments on Holographic Entanglement Entropy and RG Flows

TL;DR

This work develops a slab-based holographic c-function from entanglement entropy to study RG flows in arbitrary dimensions. It proves monotonic flow for boundary theories dual to Einstein gravity under the null energy condition, and it clarifies that in Gauss-Bonnet gravity the same entanglement-based c-function need not be monotonic due to additional curvature terms and nonlinear relations among central charges. The paper also reveals that entanglement entropy can undergo a first-order phase transition as the slab width changes, demonstrated with step and smooth bulk geometries, which induces a discontinuous change in the c-function. These results connect holographic c-theorems to central charges $C_T$ and $a_d^*$, and they outline how higher-curvature corrections may modify monotonicity properties and phase structure in holographic RG flows.

Abstract

Using holographic entanglement entropy for strip geometry, we construct a candidate for a c-function in arbitrary dimensions. For holographic theories dual to Einstein gravity, this c-function is shown to decrease monotonically along RG flows. A sufficient condition required for this monotonic flow is that the stress tensor of the matter fields driving the holographic RG flow must satisfy the null energy condition over the holographic surface used to calculate the entanglement entropy. In the case where the bulk theory is described by Gauss-Bonnet gravity, the latter condition alone is not sufficient to establish the monotonic flow of the c-function. We also observe that for certain holographic RG flows, the entanglement entropy undergoes a 'phase transition' as the size of the system grows and as a result, evolution of the c-function may exhibit a discontinuous drop.