Dynamics of Entanglement Entropy from Einstein Equation

TL;DR

The paper develops a perturbative, holographic description of how entanglement entropy responds to small metric perturbations in AdS/CFT, revealing differential equations for $ΔS_A$ that mirror Einstein dynamics in the bulk. It demonstrates, in AdS$_3$/CFT$_2$, AdS$_4$/CFT$_3$, and AdS$_5$/CFT$_4$, that $ΔS_A$ is governed by both the energy-momentum tensor $T_{tt}$ and, when present, scalar operator expectations, through nonlocal relations and, in the small subsystems limit, a universal first law-like relation with an effective temperature $T_{ m eff}=(d+1)/(2π l)$. The work also extends the notion of entanglement density to higher dimensions, showing its positivity and holographic links to boundary stress tensors, and provides explicit Green-function structures describing how local perturbations propagate entanglement. Overall, the results offer a CFT-side encoding of gravitational dynamics via entanglement and pave the way for a fuller, higher-order understanding of the gauge/gravity duality.

Abstract

We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using the AdS/CFT. This is aimed at a first step to find a counterpart of Einstein equation in the CFT language. In particular, we point out that the entanglement entropy satisfies differential equations which directly correspond to the Einstein equation in several setups of AdS/CFT. We also define a quantity called entanglement density in higher dimensional field theories and study its dynamical property for weakly excited states in conformal field theories.