Thermalization and entanglement following a non-relativistic holographic quench

TL;DR

The paper develops a holographic framework for thermalization after a quench near a quantum critical point with Lifshitz scaling ($z>1$), by generalizing the AdS-Vaidya setup to asymptotically Lifshitz spacetimes. Through geodesic and minimal-surface computations, it shows a horizon-like propagation of thermalization with a finite velocity set by the final temperature, and derives analytic upper bounds on thermalization speeds for both two-point functions and entanglement entropy. A key finding is that two-point functions thermalize with a universal velocity independent of operator dimension, while entanglement entropy thermalizes more slowly and exhibits a horizon effect similar to relativistic cases. The results illuminate non-equilibrium dynamics in non-relativistic holographic CFTs and potentially relevant finite-density systems with Lifshitz IR behavior.

Abstract

We develop a holographic model for thermalization following a quench near a quantum critical point with non-trivial dynamical critical exponent. The anti-de Sitter Vaidya null collapse geometry is generalized to asymptotically Lifshitz spacetime. Non-local 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 non-local observables.

Figures (11)

• Figure 1: Logarithm of the thermal correlator for different values of z. The different curves correspond to $z=1,2,6$, from bottom to up. 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 side figure corresponds to $z=2$ and the right hand side 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 (\ref{['eq:l']}). 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 (\ref{['eq:velocity']}).
• 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 blue surface corresponds to geodesics passing through the matter shell, while the 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 $dl/dt=2$.