Holographic Thermalization

TL;DR

This work analyzes how strongly coupled gauge theories with gravity duals thermalize after a quench by mapping energy injection to collapsing shells in AdS spacetimes and studying three nonlocal probes: equal-time two-point functions, space-like Wilson loops, and entanglement entropy. Through analytic results in AdS$_3$ and numerical analysis in AdS$_4$ and AdS$_5$, it reveals universal features: a slight delay before thermalization, a nonanalytic end to the process, and top-down (UV-first) thermalization across dimensions, with entanglement entropy often setting the equilibration timescale and saturating a causality bound. The bulk computations connect geodesics, minimal surfaces, and minimal volumes to boundary observables, demonstrating scale-dependent dynamics and occasional dimension-specific phenomena like swallow-tail transitions for certain Wilson-loop configurations. Collectively, the results illustrate how holographic models capture rapid, causality-respecting equilibration in strongly coupled plasmas and offer insights into the distinct roles of UV and IR modes in thermalization, with potential implications for QCD-like systems and heavy-ion phenomenology.

Abstract

Using the AdS/CFT correspondence, we probe the scale-dependence of thermalization in strongly coupled field theories following a quench, via calculations of two-point functions, Wilson loops and entanglement entropy in d=2,3,4. In the saddlepoint approximation these probes are computed in AdS space in terms of invariant geometric objects - geodesics, minimal surfaces and minimal volumes. Our calculations for two-dimensional field theories are analytical. In our strongly coupled setting, all probes in all dimensions share certain universal features in their thermalization: (1) a slight delay in the onset of thermalization, (2) an apparent non-analyticity at the endpoint of thermalization, (3) top-down thermalization where the UV thermalizes first. For homogeneous initial conditions the entanglement entropy thermalizes slowest, and sets a timescale for equilibration that saturates a causality bound over the range of scales studied. The growth rate of entanglement entropy density is nearly volume-independent for small volumes, but slows for larger volumes.

Figures (24)

• Figure 1: The causal structure of the Vaidya spacetime (in the $v_0 \to 0$ limit) shown in the Poincaré patch of AdS space. In this presentation, the asymptotic boundary (vertical line on the right hand side) is planar, and the null lines on the left hand side of the diagram represent the Poincaré horizon.
• Figure 2: ( A) An example space-like geodesic that starts and ends on the boundary of AdS ($z=0$) with a separation $x_0$. Outside the shell, the geodesic propagates in a black brane geometry, while inside it propagates in an empty AdS geometry. The shell refracts the geodesic. The Wightman function at scales associated to geodesics that do not penetrate the shell of matter will be thermalized. ( B) The minimal surface in AdS space associated to a circular Wilson loop. The shell of matter (indicated in light green) refracts the surface. Loops with associated surfaces that never penetrate the shell of matter will be thermalized. Both figures illustrate a quasistatic situation where the geodesic or minimal surface lies entirely at a fixed time. When the shell is dynamically falling into AdS, the geodesic or minimal surface, while remaining space-like, may not lie entirely within an equal-time surface. In both figures we are at late time when the shell is close to where the event horizon would be, so that the 'refraction' at the shell is clearly visible.
• Figure 3: The values of $E,J$ and the signs of $A_\pm,B_\pm$. The signs of $A_\pm,B_\pm$ are written in the form $(A_+\,A_-,B_+\,B_-)$. See text for the behavior of geodesics in each regime of parameters.
• Figure 4: Sample profiles of $r(\lambda)$ (in blue) in the AdS$_3$ black brane background. The event horizon $r_H =1$ is shown in red. The integration constant $\lambda_0$ appearing in \ref{['r_out(lambda)']} has been set to $\lambda_0 =- 1/4 \ln (|A_+|/|A_-|)$.
• Figure 5: $\delta {\cal L}_{\rm thermal}$ as a function of spatial scale $\ell$ for $d=2$ (red, dot dashed), $d=3$ (green) and $d=4$ (purple, dashed) for a black brane geometry with $M = 1$. The results for $d=2$ agree with the analytical results of Section \ref{['BTZanalytic']} in the limit of a shell of zero thickness.