The relation between classical and quantum Lyapunov exponent and the bound on chaos in classically chaotic quantum systems

TL;DR

This paper investigates the growth rate of out-of-time-ordered commutators in quantum chaotic systems with classical limits, revealing a universal crossover from classical to quantum chaos and demonstrating maximal scrambling consistent with the chaos bound.

Contribution

It introduces a novel approach using Wigner-Moyal expansion to connect classical Lyapunov exponents with quantum OTOC growth, showing maximal chaos in a well-defined classical-quantum system.

Findings

OTOC growth rate depends on degrees of freedom and temperature

Universal function describes classical to quantum crossover

Maximal scrambling saturates the chaos bound in deep quantum regime

Abstract

Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate $\Lambda$ as a function of the number of degrees of freedom, $f$, and inverse temperature, $\beta$. We show that the scaled growth rate, $\Lambda/f$, can be described by a universal function of $f \beta$ and displays a cross-over from classical to quantum behavior as we increase $f$ and/or lower the temperature. In the deep quantum regime of infinite $f$, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.