Fast scrambling in holographic Einstein-Podolsky-Rosen pair

TL;DR

The paper demonstrates that a holographic EPR pair, modeled by a fundamental string with oppositely accelerated endpoints in $AdS_5$, exhibits fast scrambling: a tiny perturbation of the endpoint acceleration scrambles the qubit-antiqubit correlation on a timescale $\\tau_\\ast \\sim \\beta \\ln S$ with Lyapunov exponent $\\lambda_L=\\frac{2\\pi}{\\beta}$ that saturates the chaos bound. Using a geodesic approximation on the worldsheet, which is an $AdS_2$ black hole geometry, and shock perturbations, the authors show that fast scrambling does not require an Einstein gravity dual. They also demonstrate a one-way traversable wormhole on the worldsheet when the acceleration is decreased and analyze a two-shock scenario in which the fast scrambling persists. Overall, the work clarifies that saturation of the Lyapunov bound does not guarantee a bulk Einstein gravity dual and provides a simple model for fast scrambling in a controlled setting.

Abstract

We demonstrate that a holographic model of the Einstein-Podolsky-Rosen pair exhibits fast scrambling. Strongly entangled quark and antiquark in $\mathcal{N}=4$ super Yang-Mills theory are considered. Their gravity dual is a fundamental string whose endpoints are uniformly accelerated in opposite direction. We slightly increase the acceleration of the endpoint and show that it quickly destroys the correlation between the quark and antiquark. The proper time scale of the destruction is $τ_\ast\sim β\ln S$ where $β$ is the inverse Unruh temperature and $S$ is the entropy of the accelerating quark. We also evaluate the Lyapunov exponent from correlation function as $λ_L=2π/β$, which saturates the Lyapunov bound. Our results suggest that the fast scrambling or saturation of the Lyapunov bound do not directly imply the existence of an Einstein dual. When we slightly decrease the acceleration, the quark and antiquark are causally connected and an "one-way traversable wormhole" is created on the worldsheet. It causes the divergence of the correlation function between the quark and antiquark.

Figures (6)

• Figure 1: (a) String profile of the holographic EPR pair for a fixed time slice. It is given by the semicircle whose radius is time-dependent. Endpoints of the string corresponds to the quark and antiquark. (b) Spacetime structure of the string worldsheet. Two time-like boundaries correspond to the quark and antiquark. They are causally disconnected but connected by the Einstein-Rosen bridge.
• Figure 2: Time evolution of the string profile for several fixed time slices. Parameters are set as $\tau_0=-14.51$, $a=1$ and $a'=a+10^{-6}$. Horizontal and vertical axes are proportional to $x$ and $z$ coordinates. They are normalized by $\sqrt{t^2+a^{-2}}$ to make the unperturbed string static.
• Figure 3: Spacetime structure of the perturbed string worldsheet. There is the shift $\gamma$ in origins of $(U,V)$- and $(U',V')$-coordinates.
• Figure 4: String worldsheets and geodesics between boundaries at $\tau_R=0$ and $\tau_L=0$ for $a=1$ and $a'=a+10^{-6}$. The proper times for the shock surface $\tau_0$ is chosen so that $\gamma=1, 2$ and $3$ in Figs.(a), (b) and (c), respectively.
• Figure 5: Correlation function for $\delta a>0$ and $\delta a<0$.