Evolution of Two-Point Functions from Holography

TL;DR

This work analyzes two-point functions in a 2D CFT undergoing holographic thermalization by modeling a sudden perturbation as a thin null-shell collapse in AdS3 that forms a BTZ black hole. It employs the geodesic approximation for high-dimension operators to compute boundary correlators, deriving analytic results in AdS3 and BTZ and extending them to the dynamical infalling-shell geometry. A key finding is that, for times beyond a threshold $\bar{t}$ and for separations larger than the light-cone, the correlators depend on an effective distance $l- t_1 - t_2$, while in the large-$l$ regime the correlator factorizes into a vacuum-like $l$-dependence with time-dependent thermal factors, aligning with entanglement entropy results for thermalization. The work thus links causal propagation of entangled excitations to the redistribution of energy among modes and shows how occupation-number equilibration emerges in this holographic 2D setting, highlighting regionally limited thermalization due to finite light-cone spreading. It provides a quantitative bridge between bulk geodesic lengths, boundary two-point functions, and the late-time approach to thermal equilibrium in strongly coupled, holographically dual systems.

Abstract

We consider a thermalization process in a 2-dimensional CFT that has a holographic description in terms of the gravitational collapse of a thin shell of null dust. This model represents a sudden perturbation of the CFT vacuum that communicates a uniform energy density to the system. We study the evolution of two-point functions at spacelike separated points (t1,l) and (t2,0), and reproduce the generic pattern first derived from the analysis of quantum quenches to critical systems. A crucial characteristic of these setups is that the excitations generated by the initial perturbation presents non-trivial quantum correlations. As a consequence, for any ti<infinity equilibration is only effective on finite regions whose size grows as a lightfront. The behavior on larger regions is greatly determined by the initial state, which for the quenches we consider and the holographic model has relevant differences. However in both cases for late times the dependence on the scale l of the two-point functions enters through the effective distance l-t1-t2. We interpret the onset of this behavior as an equilibration time for occupation numbers in these 2-dimensional models.

Figures (8)

• Figure 1: On the left we show the imaginary part of $t(r)$ in the different wedges of the Penrose diagram of the BTZ black hole. On the right we draw qualitatively different types of BTZ geodesics according to our classification.
• Figure 2: Example of a non equal-time BTZ geodesic completely contained in wedge I, i.e. with $\epsilon\!=\!{\tilde{\epsilon}}\!=\!1$ and $C$\ref{['CC']} positive. AdS$_3$ geodesics with $r_\ast\!>\!A$ verify the same pattern.
• Figure 3: Examples of different geodesics according to our classification in terms of $d_\text{in}$ and $d_\text{out}$. a) Geodesic with $t_1\!>\!0$, $t_2\!<\!0$ and $d_\text{in}\!=\!d_\text{out}\!=\!1$; Geodesics with $t_i\!>\!0$ and b) $d_\text{in}\!=\!d_\text{out}\!=\!1$; c) $d_\text{in}\!=\!1$ and $d_\text{out}\!=\!-1$; d) $d_\text{in}\!=\!d_\text{out}\!=\!-1$.
• Figure 4: Examples of BTZ geodesics with a) $\epsilon\!=\!1$ and $C\!<\!0$; $\epsilon\!=\!-1$ and b) $C\!<\!0$; c) $C\!>\!0$ and ${\tilde{\epsilon}}\!=\!-1$; d) $C\!>\!0$ and ${\tilde{\epsilon}}\!=\!1$, in this case $r_\ast\!<\!r_m$.
• Figure 5: For $m\!=\!1$ regularized geodesic length a) as a function of $l$ for fixed $\Delta t\!=\!3$ and $t_1\!=\!3.1,4.1,5.1$ from bottom up in solid lines ($t_2>0$), $t_1\!=\!0.9,1.9,2.9$ in dashed lines ($t_2<0$) and empty AdS geodesic in dotted line; b) as a function of $t_1$ for fixed $\Delta t\!=\!1$ and $l\!=\!2,4,6,8,10$ from bottom up respectively.