Abstract
A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results:We prove holographic bounds on the entanglement velocity vE and the butterfly effect speed vB that arises in the study of chaos.We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details of the initial state or quench protocol.In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time.We derive a bound on the rate of growth of EE for arbitrary shapes, and develop an expansion for EE at early times.In a companion paper [1], these results are put in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper.
Highlights
We obtain four key results: 1. We prove holographic bounds on the entanglement velocity vE and the butterfly effect speed vB that arises in the study of chaos
In a companion paper [1], these results are put in the broader context of entanglement entropy (EE) growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models
In a companion paper [1], we use the results of this paper along with numerical results from spin chains, to propose that the entanglement entropy in a quench in a chaotic systems is close to saturating a combination of two constraints: one that follows from recent insight into quantum chaos [25,26,27,28] and the positivity of relative entropy [29], and another bounding the rate of growth of entropy. [1] can be read as putting the results of this paper into context, relating them to the picture of the chaotic growth of operators, and analyzing them from the point of view of toy models and bounds
Summary
In a field theory model of a global quench we want to create an initial state that is shortrange entangled, translation invariant and has finite energy density. One way to create such a state is to dump in energy in an uncorrelated manner by smearing a local operator over the whole system and acting with it on the vacuum This is the setup that Liu and Suh considered holographically [19, 20]. We analyze both setups below for large regions and times R, t β, and find that the entire time evolution of entanglement entropy is universal: it does not depend on which setup we consider It has been already been demonstrated in [18,19,20] that in the linear regime (1.4) the Vaidya and the end of the world brane models give the same result.
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