Abstract
We argue that the gravitational shock wave computation used to extract the scrambling rate in strongly coupled quantum theories with a holographic dual is directly related to probing the system's hydrodynamic sound modes. The information recovered from the shock wave can be reconstructed in terms of purely diffusionlike, linearized gravitational waves at the horizon of a single-sided black hole with specific regularity-enforced imaginary values of frequency and momentum. In two-derivative bulk theories, this horizon "diffusion" can be related to late-time momentum diffusion via a simple relation, which ceases to hold in higher-derivative theories. We then show that the same values of imaginary frequency and momentum follow from a dispersion relation of a hydrodynamic sound mode. The frequency, momentum, and group velocity give the holographic Lyapunov exponent and the butterfly velocity. Moreover, at this special point along the sound dispersion relation curve, the residue of the retarded longitudinal stress-energy tensor two-point function vanishes. This establishes a direct link between a hydrodynamic sound mode at an analytically continued, imaginary momentum and the holographic butterfly effect. Furthermore, our results imply that infinitely strongly coupled, large-N_{c} holographic theories exhibit properties similar to classical dilute gases; there, late-time equilibration and early-time scrambling are also controlled by the same dynamics.
Highlights
Introduction.—The notion that dynamics at widely separated timescales is governed by independent processes lies at the heart of modern physics
We argue that the gravitational shock wave computation used to extract the scrambling rate in strongly coupled quantum theories with a holographic dual is directly related to probing the system’s hydrodynamic sound modes
We show that the same values of imaginary frequency and momentum follow from a dispersion relation of a hydrodynamic sound mode
Summary
We argue that the gravitational shock wave computation used to extract the scrambling rate in strongly coupled quantum theories with a holographic dual is directly related to probing the system’s hydrodynamic sound modes. The frequency, momentum, and group velocity give the holographic Lyapunov exponent and the butterfly velocity At this special point along the sound dispersion relation curve, the residue of the retarded longitudinal stressenergy tensor two-point function vanishes. This establishes a direct link between a hydrodynamic sound mode at an analytically continued, imaginary momentum and the holographic butterfly effect. Our results imply that infinitely strongly coupled, large-Nc holographic theories exhibit properties similar to classical dilute gases; there, late-time equilibration and early-time scrambling are controlled by the same dynamics. Ergodicity is driven by chaotic nonlinear dynamics, whereas statistics and universality drive collective behavior. Triggered by studies [10,19,20,21,22] on collective dynamics in strongly coupled large-Nc quantum systems holographically dual to black holes, Blake observed that, in the simplest such systems, 0031-9007=18=120(23)=231601(6)
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