Entanglement growth during thermalization in holographic systems

TL;DR

<3-5 sentence high-level summary> The paper analyzes the universal dynamics of nonlocal observables, esp. entanglement entropy, after global quenches in holographic CFTs by modeling the quench as a thin-shell collapse in Vaidya spacetimes. Observables are computed from the area of extremal surfaces of dimension $n$, with entanglement entropy arising from $n=d-1$, and the large-distance evolution organized into four regimes: pre-local-equilibrium quadratic growth, post-local-equilibrium linear growth, memory loss, and saturation, potentially with discontinuous or continuous transitions. A central idea is that the growth is controlled by critical extremal surfaces behind the horizon, leading to a wavefront or tsunami-like picture of entanglement propagation, and yielding universal scaling laws that depend only on horizon geometry and not on microscopic details. The results illuminate bounds on entanglement growth rates, the role of higher-derivative gravity and charges, and offer a geometric perspective on black-hole interior dynamics and holographic thermalization.

Abstract

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 CFT 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.

Figures (21)

• Figure 1: Vaidya geometry: One patches pure AdS with a black hole along an in-falling collapsing null shell located at $v=0$. We take the width of the shell to be zero which corresponds to the ${{\delta}} \mathfrak{t} =0$ limit of the boundary quench process. The spatial directions along the boundary are suppressed in the figure.
• Figure 2: Cartoon of the curve $(z_t (R, \mathfrak{t}), v_t (R, \mathfrak{t}))$ for (a) continuous and (b) discontinuous saturation. Cartoons of various extremal surfaces whose tip are labelled above are shown in Fig. \ref{['fig:surexm']}. (a): For continuous saturation the whole curve has a one-to-one correspondence to $(R,\mathfrak{t})$, and saturation happens at point $C$ continuously. (b): Discontinuous saturation happens via a jump of the extremal surface from one with tip at $C'$ to one with tip at $C$. Along the dashed portion of the curve, different points can correspond to the same $(R, \mathfrak{t})$.
• Figure 3: Cartoons of extremal surfaces with tip at various points labelled in Fig. \ref{['fig:curves']}. Spatial directions are suppressed. (a): At $\mathfrak{t}=0_+$, the extremal surface starts intersecting the null shell, with $z_c$ very small. (b) When $\mathfrak{t} \gtrsim z_h$, the extremal surface starts intersecting the null shell behind the horizon. (c) The extremal surface close to continuous saturation for which $z_t - z_c$ is small.
• Figure 4: Parametric curves $(z_t (R,\mathfrak{t}), z_c(R, \mathfrak{t}))$ at fixed $R$ and varying $\mathfrak{t}$ for Schwarzschild $h(z)$ in $d=3$. Different curves correspond to $R = 2,3, \cdots ,10$. In both plots, we choose units so that the horizon is at $z_h=1$. (a): For a strip. Note the saturation is discontinuous with $z_c$ lying behind the horizon at the saturation point where each curve stops. (b): For a sphere. The saturation is continuous and $z_c$ lies outside the horizon at the saturation point (in the plot it is too close to the horizon to be discerned).
• Figure 5: Examples of $z_m$ (blue) and $z_c^*$ (red) as functions of $z_t$ for (a): Schwarzschild $g(z)$ with $d=4$ and $n=3$, (b): RN $g(z)$ with $d=4$, $u=0.2$, and $n=3$. We have fixed $z_h=1$. Note in (b), $z_s$ does not exist and there is only region I \ref{['reg1']}.