First law of entanglement rates from holography

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. CONTENTS

Time evolution of entanglement for holographic steady state formation

“…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.…”

“…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).…”

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.…”

“…They observed a perturbative expansion for the area of the extremal surface and used that to study the post-quench evolution after an instantaneous quench. [46] used a similar method to study the very interesting case of a global quench that is linear in time.…”

“…In this subsection we will comment on the case of a linear quench g(t) ∼ t. We call this the Linear Pump. For a CFT, it was studied in detail in [46,47]. [46] in particular proposed a First Law of Entanglement Rates for small subsystems given by…”