Disrupting Entanglement of Black Holes

TL;DR

The paper links entanglement between two CFTs in a thermofield double state to the geometry of a two-sided RN-AdS black hole via the Ryu-Takayanagi prescription. It shows that mutual information between boundary regions undergoes a sharp transition controlled by temperature and region size, with a diverging critical length as extremality is approached. A generalized butterfly-effect analysis using shockwave geometries reveals a universal scrambling-timescale t_* ~ (β/2π) log(E/ΔE) (and its near-extremal Δε version), indicating that small perturbations can dramatically disrupt interboundary correlations. Together, these results illuminate how extremality, perturbations, and entanglement structure map onto the wormhole geometry, contributing to the ER=EPR dialogue and the understanding of chaotic dynamics in holographic systems.

Abstract

We study entanglement in thermofield double states of strongly coupled CFTs by analyzing two-sided Reissner-Nordstrom solutions in AdS. The central object of study is the mutual information between a pair of regions, one on each asymptotic boundary of the black hole. For large regions the mutual information is positive and for small ones it vanishes; we compute the critical length scale, which goes to infinity for extremal black holes, of the transition. We also generalize the butterfly effect of Shenker and Stanford to a wide class of charged black holes, showing that mutual information is disrupted upon perturbing the system and waiting for a time of order $\log E/δE$ in units of the temperature. We conjecture that the parametric form of this timescale is universal.

Figures (3)

• Figure 1: Values of the width $L_{\rm crit}$ of a strip for which the mutual information vanishes. We show results for black holes with different values of $R$ for the uncharged case $q=0$ (blue circles), and the near-extremal case $q=0.999\, q_{\rm ext}$ (red squares). Also shown are the approximations in Eq. (\ref{['eq-Lcrit']}). The non-(near-)extremal approximation is given by the solid (dashed) line. These plots were produced for the specific case $d=3$, $k=1$, but similar results hold for other values of $d$ and $k$.
• Figure 2: The construction of the shockwave geometry. Two halves ($L$ and $R$) are glued along the lightlike shockwave trajectory. On the right we see the result of the gluing in the $t_0\to \infty$ limit, where the net effect is a shift in the Kruskal coordinate by $\alpha$.
• Figure 3: The minimal surface (horizontal, red) in the shockwave geometry. We split the left half of the surface into three segments, labeled I, II, and III in the figure, to aid in calculation. The division between I and II occurs at the left future horizon. The smallest value of $r$ attained by the surface is $r=r_0$, which marks the division between II and III.