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arXiv:1601.00136 · Physical review. D/Physical review. D.

Holographic thermalization from nonrelativistic branes

Dibakar Roychowdhury B

Published May 31, 2016• 5 citations
arXivHTMLPDF

Abstract

In this paper, based on the fundamental principles of gauge/gravity duality and considering a global quench, we probe the physics of thermalization for certain special classes of strongly coupled nonrelativistic quantum field theories that are dual to an asymptotically Schr\"odinger $Dp$ brane space time. In our analysis, we note that during the prelocal stages of the thermal equilibrium the entanglement entropy has a faster growth in time compared to its relativistic cousin. However, it shows a linear growth during the postlocal stages of thermal equilibrium where the so-called tsunami velocity associated with the linear growth of the entanglement entropy saturates to that of its value corresponding to the relativistic scenario. Finally, we explore the saturation region and it turns out that one must constraint certain parameters of the theory in a specific way in order to have discontinuous transitions at the point of saturation.

DOI
10.1103/physrevd.93.106008

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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.
Using the holographic mapping to a gravity dual, we calculate 2-point functions, Wilson loops, and entanglement entropy in strongly coupled field theories in d=2, 3, and 4 to probe the scale dependence of thermalization following a sudden injection of energy. For homogeneous initial conditions, the entanglement entropy thermalizes slowest and sets a time scale for equilibration that saturates a causality bound. The growth rate of entanglement entropy density is nearly volume-independent for small volumes but slows for larger volumes. In this setting, the UV thermalizes first.
Motivated by exploring the thermalization process in relativistic and nonrelativistic holographic field theories after a nonlocal quench, we investigate some features in the time evolution of the entanglement wedge cross section (EWCS). This quantity is a possible holographic dual to some nonlocal information measures such as entanglement of purification. In particular, we focus on the time dependence of the EWCS during black hole formation in (D+2D+2D+2)-dimensional anti--de Sitter spacetime as well as geometries with Lifshitz and hyperscaling-violating exponents. A combination of analytic and numerical results for large symmetric strip shaped boundary subregions shows that the scaling of the EWCS at early times only depends on the Lifshitz exponent. In addition, this early-growth regime is followed by a linear-growth regime whose velocity depends on the dimensions of spacetime, the Lifshitz exponent, and the hyperscaling parameter. This velocity is the same as the entanglement velocity and for nontrivial dynamical exponent depends on the temperature of the final equilibrium state.
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 two-dimensional system with initial temperature TiT_iTi​. Subsequent equilibration increases the temperature up to a final temperature TfT_fTf​. We describe this process holographically by an injection of a thin shell of matter in the black hole background with temperature TiT_iTi​. In the limit of large regions of entanglement, the evolution of entanglement entropy at large times becomes a function of one variable ΔS(τ,ℓ)∼M(τ−ℓ)\Delta S(\tau,\ell)\sim M(\tau-\ell)ΔS(τ,ℓ)∼M(τ−ℓ). This is a so called memory loss regime well known for the holographic thermalization from zero temperature. We evaluate specific characteristics of this regime in nonequilibrium heating and find that behaviour of the function $ M(\tau-\ell)$ during post-local-equilibration and saturation times has similar forms as for thermalization from zero temperature, but the scaling coefficients depend on the temperature difference Tf−TiT_f-T_iTf​−Ti​.
Abstract We study the holographic entanglement entropy evolution after a global sharp quench of thermal state. After the quench, the system comes to equilibrium and the temperature increases fromTitoTf. Holographic dual of this process is provided by an injection of a thin shell of matter in the black hole background. The quantitative characteristics of the evolution depend substantially on the size of the initial black hole. We show that characteristic regimes during non-equilibrium heating do not depend on the initial temperature and are the same as in thermalization. Namely these regimes are pre-local-equilibration quadratic growth, linear growth and saturation regimes of the time evolution of the holographic entanglement entropy. We study the initial temperature dependence of quantitative characteristics of these regimes and find that the critical exponents do not depend on the temperature, meanwhile the prefactors are the functions on the temperature.