We develop a holographic model for thermalization following a quench near a quantum critical point with nontrivial dynamical critical exponent. The anti-de Sitter Vaidya null collapse geometry is generalized to asymptotically Lifshitz spacetime. Nonlocal observables such as two-point functions and entanglement entropy in this background then provide information about the length and time scales relevant to thermalization. The propagation of thermalization exhibits similar ``horizon'' behavior as has been seen previously in the conformal case and we give a heuristic argument for why it also appears here. Finally, analytic upper bounds are obtained for the thermalization rates of the nonlocal observables.
We develop a holographic model for thermalization following a quench near a quantum critical point with nontrivial dynamical critical exponent. The anti-de Sitter Vaidya null collapse geometry is generalized to asymptotically Lifshitz spacetime. Nonlocal observables such as two-point functions and entanglement entropy in this background then provide information about the length and time scales relevant to thermalization. The propagation of thermalization exhibits similar ``horizon'' behavior as has been seen previously in the conformal case and we give a heuristic argument for why it also appears here. Finally, analytic upper bounds are obtained for the thermalization rates of the nonlocal observables.
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+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 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 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.
The quantum null energy condition (QNEC) is a lower bound on the energy-momentum tensor in terms of the variation of the entanglement entropy of a subregion along a null direction. To gain insights into quantum thermodynamics of many-body systems, we study if the QNEC restricts irreversible entropy production in quenches driven by energy-momentum inflow from an infinite memoryless bath in two-dimensional holographic theories. We find that an increase in both entropy and temperature, as implied by the Clausius inequality of classical thermodynamics, is necessary but not sufficient to not violate QNEC in quenches leading to transitions between thermal states with momentums that are dual to Banados-Teitelboim-Zanelli geometries. For an arbitrary initial state, we can determine the lower and upper bounds on the increase of entropy (temperature) for a fixed increase in temperature (entropy). Our results provide explicit instances of quantum lower and upper bounds on irreversible entropy production whose existence has been established in literature. We also find monotonic behavior of the nonsaturation of the QNEC with time after a quench, and analytically determine their asymptotic values. Our study shows that the entanglement entropy of an interval of length l always thermalizes in time l/2 with an exponent 3/2. Furthermore, we determine the coefficient of initial quadratic growth of entanglement analytically for any l, and show that the slope of the asymptotic ballistic growth of entanglement for a semi-infinite interval is twice the difference of the entropy densities of the final and initial states. We determine explicit upper and lower bounds on these rates of growth of entanglement.
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.
Abstract We study the evolution of entanglement wedge cross section (EWCS) in the Vaidya geometry describing a thin shell of null matter collapsing into the AdS vacuum to form a black brane. In the holographic context, it is proposed that this quantity is dual to different information measures including entanglement of purification, reflected entropy, odd entropy and logarithmic negativity. In 2 + 1 dimensions, we present a combination of numerical and analytic results on the evolution and scaling of EWCS for strip shaped boundary subregions after a thermal quench. In the limit of large subregions, we find that the time evolution of EWCS is characterized by three different scaling regimes: an early time quadratic growth, an intermediate linear growth and a late time saturation. Further, in 3 + 1 dimensions, we examine the scaling behavior by considering thermal and electromagnetic quenches. In the case of a thermal quench, our numerical analysis supply results similar to observations made for the lower dimension. On the other hand, for electromagnetic quenches at zero temperature, an interesting feature is a departure from the linear behavior of the evolution to logarithmic growth.