Recent developments in the holographic description of quantum chaos

TL;DR

This work surveys the holographic description of quantum chaos, focusing on how out-of-time-ordered correlators diagnose chaos and map to bulk shock-wave dynamics near black hole horizons. The analysis establishes a universal Lyapunov bound $\lambda_L \le \frac{2\pi}{\beta}$ with black holes saturating it, and identifies the scrambling time $t_* \sim \frac{\beta}{2\pi}\log N_{\text{dof}}$ as a characteristic timescale for information spreading. It further connects chaos to entanglement spreading via mutual information and the entanglement velocity $v_E$, and links chaotic behavior to hydrodynamic transport through universal diffusion bounds involving the butterfly velocity $v_B$ and dissipation time $\tau_L$. Stringy corrections slow chaos growth but are constrained by causality, while bulk shock-wave analyses provide a coherent geometric picture of OTOC decay. Overall, the holographic perspective yields universal, testable predictions for chaotic dynamics in strongly coupled quantum systems and informs the interpretation of related transport phenomena.

Abstract

We review recent developments encompassing the description of quantum chaos in holography. We discuss the characterization of quantum chaos based on the late time vanishing of out-of-time-order correlators and explain how this is realized in the dual gravitational description. We also review the connections of chaos with the spreading of quantum entanglement and diffusion phenomena.

Figures (14)

• Figure 1: Variation of a trajectory in the phase space under small modifications of the initial condition. For a chaotic system the distance between two initially nearby trajectories increases exponentially with time, i.e., $|\delta q(t)| = |\delta q(0)| e^{\lambda t}$.
• Figure 2: Schematic form of $C(t)$. We indicated the regions of Lyapunov and Ruelle behavior. $C(t) \sim \mathcal{O}(1)$ at $t \sim t_*$.
• Figure 3: Construction of the state $W(-t)V(0)| \beta \rangle$. For a chaotic system the perturbation $V$ fails to re-materialize at $t=0$. In a non-chaotic system we expect the perturbation $V$ to re-materialize at $t=0$.
• Figure 4: Construction of the state $V(0)W(-t)| \beta \rangle$. By construction, this state displays the perturbation $V$ at $t=0$.
• Figure 5: Light cone (gray region) and butterfly effect cone (dark gray region). Inside the butterfly effect cone, for $t-t_* \geq |x|/v_B$, we have $C(t,x) \sim \mathcal{O}(1)$, whereas for outside the cone, for $t-t_* < |x|/v_B$, the commutator is small, $C(t,x) \sim 1/N_\textrm{\tiny dof} <<1$. Outside the light-cone the Lorentz invariance implies a zero commutator.