Linear response of entanglement entropy from holography

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-fromequilibrium excited states. We illustrate our findings with concrete examples, including generic power-law and periodically driven quenches.

Chaos and entanglement spreading in a non-commutative gauge theory

“…Strictly speaking, the bound (1.13) holds true for large enough subsystems. For small subsystems, the 'entanglement tsunami' picture breaks down and (1.13) can be violated instantaneously[70,71]. However, causality still implies that in average, v avg E ă 1 throughout a unitary evolution.…”

Time evolution of entanglement for holographic steady state formation

“…This is in contrast to the results of [16], where it was explicitly found in a different setup that the momentary increase rate for small regions, far away from the tsunami regime, can indeed violate the velocity bound (5.25). See also [62,63] for further discussions of entanglement entropy growth for small subsystems in different setups. A bound of the type (5.23) is especially interesting when compared to other velocities that are related to the spread of entanglement or other disturbances on the boundary of AdS d+1 , such as the entanglement velocity (5.3) and the butterfly effect velocity (5.4).…”

“…It should be pointed out that in our matching procedure of section 4, the shockwave is always assumed to be infinitely thin, hence this modification is not a result of a finite shockwave size. Also, other works where the evolution of entanglement entropy away from the tsunami regime was studied are [16,62,63], with somewhat contrasting results, as explained above.…”

Spread of Entanglement in Non-Relativistic Theories

“…In Subsection 5.3, we elaborate on the case of a global quench linear in time. This is a special case of the power law quench but this is interesting in itself due to earlier work [46,47]. We then come to Subsection 5.4 where we study a Floquet quench, a global quench that is periodic in time.…”