Entanglement Tsunami: Universal Scaling in Holographic Thermalization

TL;DR

This work considers 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.

Abstract

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 any 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.

Figures (6)

• Figure 1: Left: The growth in entanglement entropy can be visualized as occuring via an "entanglement tsunami" with a sharp wave-front carrying entanglement inward from $\Sigma$. Right: Late-time memory loss--for a wide class of compact $\Sigma$, in the limit of large size, at late times the wave front may approach that of a spherical $\Sigma$.
• Figure 2: Cartoons of the curve $(z_t (\mathfrak{t}), v_t (\mathfrak{t}))$ at fixed $\Sigma$ in a Penrose diagram for continuous (left) and discontinuous (right) saturation. The solid orange line denotes the in-falling matter shell and the red dashed line is the black hole horizon. Time progresses from bottom to top and boundary spatial directions are suppressed. Left: For continuous saturation the whole curve is single-valued throughout, and saturation happens at point $C$. Right: Discontinuous saturation happens via a jump between two branches of the curve, from $C'$ to $C$. On the dashed portion of the curve, different points $(z_t, v_t)$ can correspond to the same $\mathfrak{t}$.
• Figure 3: Left: In the pre-local-equilibration regime, the intersection of the extremal surface with the in-falling shell is close to the boundary. Right: When $\mathfrak{t} \gtrsim z_h$, the extremal surface starts intersecting with the in-falling shell behind the horizon.
• Figure 4: Upper: The dotted line denotes a curve at constant $z$, along which $v$ increases from $-\infty$ to $+\infty$ from bottom (not shown) to top. The purple line corresponds to ${{\Gamma}}_{\Sigma}^*$, the critical extremal surface, while the green lines correspond to ${{\Gamma}}_{\Sigma}$ with $v_t$ just above and below $v_t^*$. For $\mathfrak{t}$ in the linear growth regime, $z_t (\mathfrak{t})$ is very large, of order $R$, and the corresponding ${{\Gamma}}_{\Sigma}^*$ asymptotes to a constant $z=z_m$ with $z_m$ determined as described below \ref{['gene']}. Lower: At late times $z_t(\mathfrak{t})$ is $O(1)$ (i.e. does not scale with $R$), and the corresponding ${{\Gamma}}_{\Sigma}^*$ asymptotes to the horizon. This is the regime corresponding to the late time memory loss for a sphere described in \ref{['evem']}.
• Figure 5: $\mathfrak{R}_\Sigma$ for $\Sigma$ a sphere or strip, for Schwarzschild and RN black holes. We use units in which the horizon is at $z_h =1$. Upper: For $d=3$ and $\Sigma$ a sphere. The dot-dashed curves are for the Schwarzschild black hole with $R = 7$, $13$, and $50$, respectively (larger values of $\mathfrak{t}$ for the $R=13, 50$ curves are not shown due to insufficient numerics), with the top horizontal dashed line marking $v_{E}^{(\rm S)}$. Red, green, and blue curves are for the RN black hole with $(u =0.5, R=20)$, $(u=0.2, R=50)$, and $(u=0, R=50)$ respectively, and the two lower dashed horizontal lines mark $v_E$ for $u=0.5$ and $0.2$. Middle: For $d=3$ and $\Sigma$ a strip. The dot-dashed curves are for the Schwarzschild black hole with $R=7, 12, 15$. It is interesting to note their evolutions are essentially identical with the exception of different saturation times. The visible end of the dot-dashed curves coincides with discontinuous saturation for $R=7$. For $R=12$ and $15$ the curves have not been extended to saturation due to insufficient numerics. The red, green, and blue curves are for the RN black hole with $(u=0.5, R=5)$, $(u=0.5, R=6)$, and $(u=0, R=6)$, respectively. The $u=0.5$ curve ends at saturation, but for $u=0.2$ and $0$, saturation happens at larger values of $\mathfrak{t}$ than shown. Lower: For $d=4$ and $\Sigma$ a sphere. The color and pattern scheme is identical to the upper plot, but the Schwarzschild curves are at $R=7$, $12$, and $50$, respectively, and $u=0.5$, $0.2$, $0$ curves are all at $R=20$.