Thermalization of Strongly Coupled Field Theories

TL;DR

Using the holographic mapping to a gravity dual, 2-point functions, Wilson loops, and entanglement entropy in strongly coupled field theories in d=2, 3, and 4 are calculated to probe the scale dependence of thermalization following a sudden injection of energy.

Abstract

Using the AdS/CFT correspondence, we probe the scale-dependence of thermalization in strongly coupled field theories following a quench via saddlepoint calculations of 2-point functions, Wilson loops and entanglement entropy in $d=2,3,4$. For homogeneous initial conditions, the entanglement entropy thermalizes slowest, and sets a timescale for equilibration that saturates a causality bound. The growth rate of entanglement entropy density is nearly volume-independent for small volumes, but slows for larger volumes. In this strongly coupled setting, the UV thermalizes first.

Figures (5)

• Figure 1: $\delta \tilde{\cal L} - \delta \tilde{\cal L}_{{\rm thermal}}$ ($\tilde{\cal L} \equiv {\cal L} / \ell$) as a function of boundary time $t_0$ for $d=2,3,4$ (left,right, middle) for a thin shell ($v_0 = 0.01$). The boundary separations are $\ell = 1,2,3,4$ (top to bottom curve). All quantities are given in units of $M$. These numerical results match analytical results for $d=2$ as $v_0 \to 0$.
• Figure 2: Thermalization times ($\tau_{{\rm dur}}$, top line; $\tau_{{\rm max}}$, middle line; $\tau_{1/2}$, bottom line) as a function of spatial scale for $d=2$ (left), $d=3$ (middle) and $d=4$ (right) for a thin shell ($v_0 = 0.01$). All thermalization time scales are linear in $\ell$ for $d=2$, and deviate from linearity for $d=3,4$.
• Figure 3: $\delta \tilde{\cal A} - \delta \tilde{\cal A}_{{\rm thermal}}$ ($\tilde{\cal A} \equiv {\cal A}/\pi R^2$; left and middle panels) and $\delta \tilde{ V} - \delta \tilde{ V}_{{\rm thermal}}$ ($\tilde{V} \equiv V/(4 \, \pi R^3 / 3)$; right panel) as a function of $t_0$ for radii $R = 0.5, 1, 1.5, 2$ (top curve to bottom curve) and mass shell parameters $v_0 = 0.01$, $M=1$, in $d=3$ (left panel) and $d=4$ (middle and right panel) field theories.
• Figure 4: Thermalization times ($\tau_{{\rm dur}}$, top line; $\tau_{{\rm max}}$, middle line; $\tau_{1/2}$, bottom line) as a function of the diameter for circular Wilson loops in $d=3,4$ (left, middle) and for entanglement entropy of spherical regions in $d=4$ (right).
• Figure 5: (Left) Maximal growth rate of entanglement entropy density vs. radius of entangled region for $d=2,3,4$ (top to bottom). (Middle) Same plot for $d=2$, larger range of $\ell$. (Right) Maximal entropy growth rate for $d=2$.