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

• URL: http://arxiv.org/abs/2512.19869v1
• Date: Mon, 22 Dec 2025 20:52:17 GMT
• Title: The relation between classical and quantum Lyapunov exponent and the bound on chaos in classically chaotic quantum systems
• Authors: Fabian Haneder, Gerrit Caspari, Juan Diego Urbina, Klaus Richter,
• Abstract summary: We provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit.<n>We show that the scaled growth rate, $/f$, can be described by a universal function of $f $.<n>In the deep quantum regime of infinite $f$, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos.
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• License: http://creativecommons.org/licenses/by/4.0/
• 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 $Λ$ as a function of the number of degrees of freedom, $f$, and inverse temperature, $β$. We show that the scaled growth rate, $Λ/f$, can be described by a universal function of $f β$ 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.

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