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arXiv:2001.03172 · Physical Review Letters

Black Holes Often Saturate Entanglement Entropy the Fastest

Márk Mezei, Wilke van der Schee

Published May 20, 2020• 12 citations
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Abstract

There is a simple bound on how fast the entanglement entropy of a subregion of a many-body quantum system can saturate in a quench: t_{sat}≥R/v_{B}, where t_{sat} is the saturation time, R the radius of the largest inscribed sphere, and v_{B} the butterfly velocity characterizing operator growth. By combining analytic and numerical approaches, we show that in systems with a holographic dual, the saturation time is equal to this lower bound for a variety of differently shaped entangling surfaces, implying that the dual black holes saturate the entanglement entropy as fast as possible. This finding adds to the growing list of tasks that black holes are the fastest at. We furthermore analyze the complete time evolution of entanglement entropy for large regions with a variety of shapes, yielding more detailed information about the process of thermalization in these systems.

DOI
10.1103/physrevlett.124.201601

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There is a simple bound on how fast the entanglement entropy of a subregion of a many-body quantum system can saturate in a quench: t_{sat}≥R/v_{B}, where t_{sat} is the saturation time, R the radius of the largest inscribed sphere, and v_{B} the butterfly velocity characterizing operator growth. By combining analytic and numerical approaches, we show that in systems with a holographic dual, the saturation time is equal to this lower bound for a variety of differently shaped entangling surfaces, implying that the dual black holes saturate the entanglement entropy as fast as possible. This finding adds to the growing list of tasks that black holes are the fastest at. We furthermore analyze the complete time evolution of entanglement entropy for large regions with a variety of shapes, yielding more detailed information about the process of thermalization in these systems.
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: 1. We prove holographic bounds on the entanglement velocity vE and the butterfly effect speed vB that arises in the study of chaos. 2. 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. 3. In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time. 4. 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, 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.
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: 1. We prove holographic bounds on the entanglement velocity v E and the butterfly effect speed v B that arises in the study of chaos. 2. 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. 3. In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time. 4. 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.
We derive in detail several universal features in the time evolution of entanglement entropy and other nonlocal observables in quenched holographic systems. The quenches are such that a spatially uniform density of energy is injected at an instant in time, exciting a strongly coupled conformal field theory which eventually equilibrates. Such quench processes are described on the gravity side by the gravitational collapse of a thin shell that results in a black hole. Various nonlocal observables have a unified description in terms of the area of extremal surfaces of different dimensions. In the large distance limit, the evolution of an extremal surface, and thus the corresponding boundary observable, is controlled by the geometry around and inside the event horizon of the black hole, allowing us to identify regimes of pre-local-equilibration quadratic growth, post-local-equilibration linear growth, a memory loss regime, and a saturation regime with behavior resembling those in phase transitions. We also discuss possible bounds on the maximal rate of entanglement growth in relativistic systems.
We consider the time evolution of entanglement entropy after a global quench in a strongly coupled holographic system, whose subsequent equilibration is described in the gravity dual by the gravitational collapse of a thin shell of matter resulting in a black hole. In the limit of large regions of entanglement, the evolution of entanglement entropy is controlled by the geometry around and inside the event horizon of the black hole, resulting in regimes of pre-local-equilibration quadratic growth (in time), post-local-equilibration linear growth, a late-time regime in which the evolution does not carry memory of the size and shape of the entangled region, and a saturation regime with critical behavior resembling those in continuous phase transitions. Collectively, these regimes suggest a picture of entanglement growth in which an "entanglement tsunami" carries entanglement inward from the boundary. We also make a conjecture on the maximal rate of entanglement growth in relativistic systems.
We study the evolution and scaling of the entanglement entropy after two types of quenches for a 2+1 field theory, using a conjectured holographic technique for its computation. We study a thermal quench, dual to the addition of a shell of uncharged matter to four-dimensional anti-de Sitter (AdS 4 ) spacetime, and study the subsequent formation of a Schwarzschild black hole. We also study an electromagnetic quench, dual to the addition of a shell of charged sources to AdS 4 , following the subsequent formation of an extremal dyonic black hole. In these backgrounds, we consider the entanglement entropy of two types of geometries, the infinite strip and the round disc, and find distinct behavior for each. Some of our findings naturally supply results analogous to observations made in the literature for lower dimensions, but we also uncover several new phenomena, such as (in some cases) a discontinuity in the time derivative of the entanglement entropy as it nears saturation, and for the electromagnetic quench, a logarithmic growth in the entanglement entropy with time for both the disc and strip, before settling to saturation. We briefly discuss the possible origin of the new phenomena in terms of the features of the conjectured dual field theory.
We calculate the evolution of the geometric entanglement entropy following a local quench in the D1-D5 conformal field theory, a two-dimensional theory that describes a particular bound state of D1 and D5 branes. The quench corresponds to a localized insertion of the exactly marginal operator that deforms the field theory off of the orbifold (free) point in its moduli space. This deformation ultimately leads to thermalization of the system. We find an exact analytic expression for the entanglement entropy of any spatial interval as a function of time after the quench and analyze its properties. This process is holographically dual to one stage in the formation of a stringy black hole.