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Memory Loss in Holographic Non-equilibrium Heating

Dmitry Sergeevich Ageev, I. Ya. Aref’eva

Published Apr 25, 2017• 7 citations
OpenAlex

Abstract

We consider the time evolution of entanglement entropy after a global quench in a strongly coupled holographic two-dimensional system with initial temperature $T_i$. Subsequent equilibration increases the temperature up to a final temperature $T_f$. We describe this process holographically by an injection of a thin shell of matter in the black hole background with temperature $T_i$. In the limit of large regions of entanglement, the evolution of entanglement entropy at large times becomes a function of one variable $\Delta S(\tau,\ell)\sim M(\tau-\ell)$. 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 $T_f-T_i$.

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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(τ−ℓ) M(\tau-\ell)M(τ−ℓ) 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.
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 study thermalization in the holographic (1 + 1)-dimensional CFT after simultaneous generation of two high-energy excitations in the antipodal points on the circle. The holographic picture of such quantum quench is the creation of BTZ black hole from a collision of two massless particles. We perform holographic computation of entanglement entropy and mutual information in the boundary theory and analyze their evolution with time. We show that equilibration of the entanglement in the regions which contained one of the initial excitations is generally similar to that in other holographic quench models, but with some important distinctions. We observe that entanglement propagates along a sharp effective light cone from the points of initial excitations on the boundary. The characteristics of entanglement propagation in the global quench models such as entanglement velocity and the light cone velocity also have a meaning in the bilocal quench scenario. We also observe the loss of memory about the initial state during the equilibration process. We find that the memory loss reflects on the time behavior of the entanglement similarly to the global quench case, and it is related to the universal linear growth of entanglement, which comes from the interior of the forming black hole. We also analyze general two-point correlation functions in the framework of the geodesic approximation, focusing on the study of the late time behavior.
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 holographically the out of equilibrium dynamics of a finite size closed quantum system in 2 + 1 dimensions, modelled by the collapse of a shell of a massless scalar field in AdS4. In global coordinates there exists a variety of evolutions towards final black hole formation which we relate with different patterns of relaxation in the dual field theory. For large scalar initial data rapid thermalization is achieved as a priori expected. Interesting phenomena appear for small enough amplitudes. Such shells do not generate a black hole by direct collapse, but quite generically, an apparent horizon emerges after enough bounces off the AdS boundary. We relate this bulk evolution with relaxation processes at strong coupling which delay in reaching an ergodic stage. Besides the dynamics of bulk fields, we monitor the entanglement entropy, finding that it oscillates quasi-periodically before final equilibration. The radial position of the travelling shell is brought in correspondence with the evolution of the pattern of entanglement in the dual field theory. We propose, thereafter, that the observed oscillations are the dual counterpart of the quantum revivals studied in the literature. The entanglement entropy is not only able to portrait the streaming of entangled excitations, but it is also a useful probe of interaction effects.
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.