A First Law of Entanglement Rates from Holography

TL;DR

This paper investigates the evolution of entanglement entropy in holographic CFTs under time-dependent perturbations, establishing a first law for the rates of change of entanglement entropy and energy.

Contribution

It demonstrates that while the traditional first law of entanglement entropy may be violated, a first law for the rates of entanglement and energy changes still holds in holography.

Findings

First law for rates of entanglement entropy and energy holds in holography.

Traditional first law may be violated under certain time-dependent perturbations.

First law for rates applies to any asymptotic AdS metric perturbation with specific t-dependence.

Abstract

For a perturbation of the state of a Conformal Field Theory (CFT), the response of the entanglement entropy is governed by the so-called "first law" of entanglement entropy, in which the change in entanglement entropy is proportional to the change in energy. Whether such a first law holds for other types of perturbations, such as a change to the CFT Lagrangian, remains an open question. We use holography to study the evolution in time $t$ of entanglement entropy for a CFT driven by a $t$-linear source for a conserved $U(1)$ current or marginal scalar operator. We find that although the usual first law of entanglement entropy may be violated, a first law for the rates of change of entanglement entropy and energy still holds. More generally, we prove that this first law for rates holds in holography for any asymptotically $(d+1)$-dimensional Anti-de Sitter metric perturbation whose $t$ dependence first appears at order $z^d$ in the Fefferman-Graham expansion about the boundary at $z=0$.