Thermalization of mutual and tripartite information in strongly coupled two dimensional conformal field theories

TL;DR

The paper investigates how mutual and tripartite information thermalize in strongly coupled two-dimensional conformal field theories using holography. Energy injection is modeled by a Vaidya-AdS3 background, and entanglement entropies are computed from geodesic lengths to obtain I(A,B) and I3(A,B,C) for various interval configurations. Mutual information generically rises from the vacuum value to a peak before decaying to the thermal value, with the peak timing depending on interval sizes and separations; a simple causality-based picture explains much of this behavior. Tripartite information remains nonzero and time-dependent during the process and is typically non-positive, contrasting with some quenched, large-time results in the literature and underscoring the importance of initial long-range correlations in dynamical holographic setups.

Abstract

The mutual and tripartite information between pairs and triples of disjoint regions in a quantum field theory are sensitive probes of the spread of correlations in an equilibrating system. We compute these quantities in strongly coupled two-dimensional conformal field theories with a gravity dual following the homogenous deposition of energy. The injected energy is modeled in anti-de Sitter space as an infalling shell, and the information shared by disjoint intervals is computed in terms of geodesic lengths in this background. For given widths and separation of the intervals, the mutual information typically starts at its vacuum value, then increases in time to reach a maximum, and then declines to the value at thermal equilibrium. A simple causality argument qualitatively explains this behavior. The tripartite information is generically non-zero and time-dependent throughout the process. This contrasts with (but does not contradict) the time-independent tripartite information one finds after a two-dimensional quantum quench in the limit of large time and distance scales compared to the initial inverse mass gap.

Figures (8)

• Figure 1: ( a) The causal structure of the Vaidya spacetime shown in the Poincaré patch of AdS space. 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. ( b) Connected (in blue) and disconnected (in red dashed) locally minimal surfaces for the boundary region $A \cup B$ in AdS$_3$.
• Figure 2: $d_{\text{thermal}}$ and $d_{\text{vacuum}}$ as a function of $\ell$ for $r_H=1$. The mutual information surely vanishes for $t_0 \le 0$ and $t_0 \ge (2\ell+d)/2$ for $d$ and $\ell$ in the white region, for $t_0 \ge (2\ell+d)/2$ in the green shaded region, while it is generically everywhere non-zero in the blue dotted region.
• Figure 3: Rescaled mutual information $\tilde{I} \equiv 4 G_N I$ as a function of boundary time $t_0$ for $d=0.4$ (left), $d=2$ (center) and $d=4$ (right), and $r_H =1$. The various curves correspond to different values of $\ell$ which increases from the bottom up. The left panel shows $\ell = 0.2, 0.4, \dots , 2.0$, while the center panel and the right one show $\ell = 1,2,\dots,10$ (some of the curves in the three panels are not visible because everywhere vanishing).
• Figure 4: Rescaled mutual information $\tilde{I} \equiv 4 G_N I$ as a function of boundary time $t_0$ for $d=0.4$, $\ell_1 = 2$ (left), $d=2$, $\ell_1 = 8$ (center) and $d=4$, $\ell_1 =8$ (right), and $r_H =1$. The various curves correspond to different values of $\ell_2$ which increases from the bottom up. The left panel shows $\ell_2 = 0.4, 0.8, \dots, 4.0$ while the center and the right panels have $\ell_2 = 2,4, \dots, 20$.
• Figure 5: ( a) A simple causality derived picture for two intervals after a quench. After the quench at $t=0$ signals propagate outwards at the speed of light in both directions. These will give correlations which contribute to the mutual information at $t=t_0$ if one signal is in each interval, which occurs only for signals originating in the shaded pink diamond. ( b) The analogous picture for two intervals of different length.