Scrambling time from local perturbations of the eternal BTZ black hole

TL;DR

The paper analyzes scrambling of correlations in two 2d CFTs at large central charge in the thermofield double setup after a local perturbation. It combines a CFT computation of mutual information via twist operators and a holographic calculation using a bulk BTZ black hole with a free-falling particle to derive the scrambling time $t^\star_\omega$, showing exact agreement between field theory and gravity for all $t_\omega$. The work reveals a universal fast-scrambling behavior, with a logarithmic dependence on perturbation energy in the late-time regime, and connects the boundary dynamics to bulk shock-wave propagation. The results provide a concrete, analytic demonstration of scrambling in holographic theories and deepen the understanding of how local excitations destroy preexisting entanglement in the thermofield double framework.

Abstract

We compute the mutual information between finite intervals in two non-compact 2d CFTs in the thermofield double formulation after one of them has been locally perturbed by a primary operator at some time $t_ω$ in the large $c$ limit. We determine the time scale, called the scrambling time, at which the mutual information vanishes and the original entanglement between the thermofield double gets destroyed by the perturbation. We provide a holographic description in terms of a free falling particle in the eternal BTZ black hole that exactly matches our CFT calculations. Our results hold for any time $t_ω$. In particular, when the latter is large, they reproduce the bulk shock-wave propagation along the BTZ horizon description.

Figures (4)

• Figure 1: Our setup in the computation of the mutual information. We have two intervals $A$ and $B$ of size $L_2-L_1=L$ in each CFT and a local operator inserted at time $t_w$ in the past. The operators are separated by distance $2\epsilon$ and in the CFT formulas we use $L_1=y$ and $L_2=y+L$.
• Figure 2: Plot shows our time-like geodesic on Kruskal diagram. The red part is given by \ref{['TXL']} and the full geodesic (blue) by \ref{['TF']}. Plot for $M=10$, $\epsilon=0.01$ and $t_\omega=0.25$
• Figure 3: In the left picture (the second choice \ref{['order2']}), one returns to the starting point after going around two standard cylinders. In the right picture (the first choice \ref{['order1']}), this operation involves going around $n$ standard cylinders.
• Figure 4: In the replica geometry which we obtain by gluing n cylinders along the cuts, we can go around 2 standard cylinders, for example, $i$-th and $(i+1)$-th cylinders (dotted line).