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

TL;DR

Problem: relate classical chaotic dynamics to quantum scrambling in many-degree systems and test the MSS bound in a well-defined quantum chaotic model with a classical limit. Approach: develop a semiclassical Wigner–Moyal quantization framework and apply ensemble-equivalence-based saddle-point analysis to a high-dimensional hyperbolic manifold with known DOS and Lyapunov exponent. Key results: leading-order OTOC growth yields a canonical quantum Lyapunov exponent Λ that saturates the MSS bound in the f→∞ limit and exhibits a universal Λ( fβ ) scaling; subleading ħ^2 corrections are analyzed and connected to gravity duals like JT gravity. Impact: provides first evidence of maximal scrambling in a unitary quantum chaotic system with a clear classical limit and offers a controlled route to gravity–duality interpretations without disorder averaging.

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.

Figures (4)

• Figure 1: The OTOC $\mathcal{C}(t)=\ev{\abs{[\hat{n}_1(t),\hat{n}]}^2}_\psi$ in a strongly kicked Bose-Hubbard system with $N=10000$ particles and $L=2$ sites, where $\hat{n}_1$ is the particle number operator on the first site and $\psi$ is a suitably localized coherent state at energy $E$. Left: early times. The nearly linear slope indicates the initial exponential growth with a constant rate corresponding to twice the classical Lyapunov exponent. Right: The OTOC saturates at late times. Blue solid line: numerical data. Orange opaque line: linear fit. Taken with permission from M. Steinhuber steinhuber.
• Figure 2: Example surface for a Hadamard-Gutzwiller-like model. The blue curve is a possible periodic orbit of the classical dynamics. In order for the motion to be chaotic, the surface has to have genus at least 2. Adapted from haneder_beyond_2025.
• Figure 3: Quantum Lyapunov exponent $\Lambda=2\lambda(E_\beta^*)$ for inverse temperature $\beta=1$, evaluated for different dimensions $f$, plotted in units where $\hbar=1,m=2,L=1,\lambda_{\mathrm{th}}=1/2$. Solid blue line: full numerical solution of the stationarity condition \ref{['eq:stationarity']}. Dotted orange line: first correction to the infinite-dimensional solution for large, finite dimension \ref{['eq:large-dim-lyapunov']}. Grey dashed line: $2\pi$, i.e. the MSS bound, eq. (\ref{['eq:mss-bound']}). One can see that both finite-dimensional results approach the MSS bound for (very) large dimensionality.
• Figure 4: Double-$\log$ plot of the system's rescaled leading-order quantum Lyapunov exponent $(1/f)\Lambda$ (where $\Lambda=2\lambda(E_\beta^*)$) as a function of the rescaled inverse temperature $f\beta$. The slightly opaque solid lines show $(1/f)\Lambda(f\beta)$, obtained from the full numerical solution of the stationarity condition \ref{['eq:stationarity']} for two representative dimensions, $f\!=\!301$ and $f\!=\!4301$, in units where $\hbar=1,m=2,L=1$. The two numerical curves are so close to each other that their difference cannot be resolved in the plot. This indicates $(f\beta)$-scale invariance in the large-$f$ limit. Dotted orange line: classical approximation to the Lyapunov exponent \ref{['eq:small-dim-lyapunov']}, given by the equipartition theorem. Dashed grey line: $2\pi/\beta$, i.e. the MSS bound eq. (\ref{['eq:infinite-lyapunov']}). The system saturates the MSS bound at low (scaled) temperatures, while it is more appropriately described by classical equipartition (eq. \ref{['eq:small-dim-lyapunov']}) at high (scaled) temperatures.