Thermalization and entanglement following a nonrelativistic holographic quench

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.

Figure 1

Logarithm of the thermal correlator for different values of z. The different curves correspond to $z=1$, 2, 6, from bottom to top. The (red) line is a line with slope $-1$. The figure shows that the thermal correlation function is fairly independent of the value of $z$.

Figure 2

In both of the figures the dot-dashed curve (red) corresponds to the apparent horizon, the dashed curve (blue) to the event horizon, and the solid curve (black) to a generic geodesic contributing to the two-point function. The left-hand figure corresponds to $z=2$ and the right-hand figure to $z=3$. The geodesic is seen to pass through both horizons, once through the event horizon and twice through the apparent horizon.

Figure 3

Lower bounds for thermalization times for different values of $z$. The data points are obtained by numerical integration of the integral in (49). From top to bottom the different colored data points correspond to the values $z=1$, 2, 3, 4, 5. The lines with the corresponding colors are best fit lines with slopes given by (54).

Figure 4

Logarithm of the quench correlator for $z=1$, with the vacuum value subtracted. The (red) line corresponds to twice the speed of light. The narrow (blue) surface corresponds to geodesics passing through the matter shell, while the wide (green) surface corresponds to geodesics probing the $u<1$ region.

Figure 5

Logarithm of the quench correlator for $z=2$, with the vacuum value subtracted. The (red) line is a reference line with slope $dl/dt=2$.

Figure 6

(left-hand figure) Logarithm of the quench correlator for $z=3$ with the vacuum value subtracted. The (red) line is a reference line with slope $dl/dt=2$. The thermalization is seen to happen very suddenly as there are several geodesics contributing. (right-hand figure) The (blue) solid line is the real thermalization time extracted from the correlator, while the (red) dashed line is a reference line with slope 1 and the (brown) dot-dashed line is the lower bound for the thermalization time as obtained from numerical integration of (48, 49).

Figure 7

Logarithm of the two-point correlator, with its thermal value subtracted. The different curves correspond to different values of $l$ as $l=4$, 8, 12 from bottom to top. The left-hand figure is for $z=2$ while the right-hand figure is for $z=3$.

Figure 8

The finite part of the entanglement entropy density as a function of the width of the strip $l$, in the thermal state. The different curves correspond to the values $z=2$, 4, 6 from top to bottom. The straight line is a reference line with slope 1.

Figure 9

(left-hand figure) The entanglement entropy density in the quench state as a function of the width of the strip $l$ and the time $t$, for the case of $z=2$. The (red) line corresponds to the upper bound velocity, which in dimensionless coordinates is simply 1. (right-hand figure) The (blue) solid curve is the real thermalization time, while the (brown) dot-dashed curve is the lower bound for the thermalization time and the (red) dashed curve is a reference line with slope 1. As the figure shows, thermalization of the entanglement entropy really happens more slowly than the upper bound due to multiple surfaces contributing to the same values of $(l,t)$.

Figure 10

Evolution of the entanglement entropy density in the quench state for $z=2$, with the thermal value subtracted. The different curves correspond to the values $l=4$, 8, 12.

Figure 11

(left-hand figure) The logarithm of the two-point correlation function with vacuum value subtracted, for $z=2$ and $3+1$ dimensional dual field theory. The (red) curve corresponds to the upper bound speed $v=\sqrt{10}$ from (a4). (right-hand figure) The entanglement entropy density for the same $d$ and $z$. The (red) curve corresponds to the upper bound speed $v=\sqrt{10/3}$ from (a10).