# Linear response of entanglement entropy from holography

## TL;DR

This paper investigates how entanglement entropy responds to global quenches in holographic models, revealing a linear response relation that generalizes the first law of entanglement and introduces a new order parameter for far-from-equilibrium states.

## Contribution

It introduces a linear response framework for entanglement entropy in holography, extending the first law to dynamic quenches and proposing a new measure for thermalization.

## Key findings

- Linear response replaces the first law for rapid quenches.
- An extra term resembling a time-dependent relative entropy appears during thermalization.
- The framework applies to power-law and periodically driven quenches.

## Abstract

For time-independent excited states in conformal field theories, the entanglement entropy of small subsystems satisfies a `first law'-like relation, in which the change in entanglement is proportional to the energy within the entangling region. Such a law holds for time-dependent scenarios as long as the state is perturbatively close to the vacuum, but is not expected otherwise. In this paper we use holography to investigate the spread of entanglement entropy for unitary evolutions of special physical interest, the so-called global quenches. We model these using AdS-Vaidya geometries. We find that the first law of entanglement is replaced by a linear response relation, in which the energy density takes the role of the source and is integrated against a time-dependent kernel with compact support. For adiabatic quenches the standard first law is recovered, while for rapid quenches the linear response includes an extra term that encodes the process of thermalization. This extra term has properties that resemble a time-dependent `relative entropy'. We propose that this quantity serves as a useful order parameter to characterize far-from-equilibrium excited states. We illustrate our findings with concrete examples, including generic power-law and periodically driven quenches.