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arXiv:2610.15880

From Lyapunov Growth to Weak Quantum Chaos

Sašo Grozdanov, Tomaž Prosen

Published Oct 11, 2026• 0 citations
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Abstract

Local out-of-time-order correlators (OTOCs) measure the growth of initially small commutators. For bounded local operators, this growth ends when the local commutator becomes of order one, but in an extended system the region over which it is large can continue to expand. We study this spatial proliferation using three related observables: the local squared commutator $C_r(t)$, its spatial sum $\bar C(t) = \sum_r C_r(t)$, and the symmetrically regularised connected dOTOC $D(t)$ of two extensive observables. We show that Lieb--Robinson locality gives $\bar C(t) \lesssim t^d$. We then prove that, in a Gibbs state with uniform exponential clustering, the full dOTOC obeys the same bound, $D(t) \lesssim t^d$, improving our earlier suboptimal $t^{3d}$ bound. We illustrate these results in two quantum many-body systems: a Brownian coupled-cluster model and a finite-temperature SYK chain. In both cases, an exact disorder symmetry relates $D(t)$ to $\bar C(t)$ and allows us to follow the same extensive observable from the pre-scrambling Lyapunov regime into the post-scrambling growth associated with an expanding saturated region. This late-time power-law growth is the concrete realisation of the weak quantum chaos introduced in our earlier work.

DOI
10.48550/arxiv.2610.15880

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Local out-of-time-order correlators (OTOCs) measure the growth of initially small commutators. For bounded local operators, this growth ends when the local commutator becomes of order one, but in an extended system the region over which it is large can continue to expand. We study this spatial proliferation using three related observables: the local squared commutator Cr(t)C_r(t)Cr​(t), its spatial sum Cˉ(t)=∑rCr(t)\bar C(t) = \sum_r C_r(t)Cˉ(t)=∑r​Cr​(t), and the symmetrically regularised connected dOTOC D(t)D(t)D(t) of two extensive observables. We show that Lieb--Robinson locality gives Cˉ(t)≲td\bar C(t) \lesssim t^dCˉ(t)≲td. We then prove that, in a Gibbs state with uniform exponential clustering, the full dOTOC obeys the same bound, D(t)≲tdD(t) \lesssim t^dD(t)≲td, improving our earlier suboptimal t3dt^{3d}t3d bound. We illustrate these results in two quantum many-body systems: a Brownian coupled-cluster model and a finite-temperature SYK chain. In both cases, an exact disorder symmetry relates D(t)D(t)D(t) to Cˉ(t)\bar C(t)Cˉ(t) and allows us to follow the same extensive observable from the pre-scrambling Lyapunov regime into the post-scrambling growth associated with an expanding saturated region. This late-time power-law growth is the concrete realisation of the weak quantum chaos introduced in our earlier work.
Out-of-time-ordered correlation functions (OTOCs) are presently being extensively debated as quantifiers of dynamical chaos in interacting quantum many-body systems. We argue that in quantum spin and fermionic systems, where all local operators are bounded, an OTOC of local observables is bounded as well and thus its exponential growth is merely transient. As a better measure of quantum chaos in such systems, we propose, and study, the density of the OTOC of extensive sums of local observables, which can exhibit indefinite growth in the thermodynamic limit. We demonstrate this for the kicked quantum Ising model by using large-scale numerical results and an analytic solution in the integrable regime. In a generic case, we observe the growth of the OTOC density to be linear in time. We prove that this density in general, locally interacting, nonintegrable quantum spin and fermionic dynamical systems exhibits growth that is at most polynomial in time---a phenomenon, which we term weak quantum chaos. In the special case of the model being integrable and the observables under consideration quadratic, the OTOC density saturates to a plateau.
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 ΛΛΛ as a function of the number of degrees of freedom, fff, and inverse temperature, βββ. We show that the scaled growth rate, Λ/fΛ/fΛ/f, can be described by a universal function of fβf βfβ and displays a cross-over from classical to quantum behavior as we increase fff and/or lower the temperature. In the deep quantum regime of infinite fff, 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.
Scrambling of information in a quantum many-body system, quantified by the out-of-time-ordered correlator (OTOC), is a key manifestation of quantum chaos. A regime of exponential growth in the OTOC, characterized by a Lyapunov exponent, has so far mostly been observed in systems with a high-dimensional local Hilbert space and in weakly-coupled systems. Here, we propose a general criterion for the existence of a well-defined regime of exponential growth of the OTOC in spatially extended systems with local interactions. In such systems, we show that a parametrically long period of exponential growth requires the butterfly velocity to be much larger than the Lyapunov exponent times a microscopic length scale, such as the lattice spacing. As an explicit example, we study a random unitary circuit with tunable interactions. In this model, we show that in the weakly interacting limit, the above criterion is satisfied, and there is a prolonged window of exponential growth. Our results are based on numerical simulations of both Clifford and universal random circuits supported by an analytical treatment.
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 fβ 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.
Out-of-time-order correlators (OTOCs), vigorously being explored as a measure of quantum chaos and information scrambling, is studied here in the natural and simplest multiparticle context of bipartite systems. We show that two strongly chaotic and weakly interacting subsystems display two distinct regimes in the growth of OTOCs. The first is dominated by intrasubsystem scrambling, when an exponential growth with a positive Lyapunov exponent is observed until the Ehrenfest time. This regime is essentially independent of the interaction, while the second one is an interaction dominated exponential approach to saturation that is universal and described by a random matrix model. This simple random matrix model of weakly interacting strongly chaotic bipartite systems, previously employed for studying entanglement and spectral transitions, is approximately analytically solvable for its OTOC. The example of two coupled kicked rotors is used to demonstrate the different regimes, and the extent to which the random matrix model is applicable. We remark that the second regime implies an emergent invariance of the OTOC under local unitary transformations suggesting that the rate of relaxation is connected to the entangling power of the interaction. That the two regimes correspond to delocalization in the subsystems followed by intersubsystem mixing is seen via the participation ratio in phase space. We also point out that the second, universal, regime alone exists when the observables are in a sense locally prescrambled.
Fast scrambling, quantified by the exponential initial growth of out-of-time-ordered correlators (OTOCs), is the ability to efficiently spread quantum correlations among the degrees of freedom of interacting systems and constitutes a characteristic signature of local unstable dynamics. As such, it may equally manifest both in systems displaying chaos or in integrable systems around criticality. Here we go beyond these extreme regimes with an exhaustive study of the interplay between local criticality and chaos right at the intricate phase-space region where the integrability-chaos transition first appears. We address systems with a well-defined classical (mean-field) limit, as coupled large spins and Bose-Hubbard chains, thus allowing for semiclassical analysis. Our aim is to investigate the dependence of the exponential growth of the OTOCs, defining the quantum Lyapunov exponent λ_{q} on quantities derived from the classical system with mixed phase space, specifically the local stability exponent of a fixed point λ_{loc} as well as the maximal Lyapunov exponent λ_{L} of the chaotic region around it. By extensive numerical simulations covering a wide range of parameters we give support to a conjectured linear dependence 2λ_{q}=aλ_{L}+bλ_{loc}, providing a simple route to characterize scrambling at the border between chaos and integrability.