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arXiv:2109.03826 · Journal of High Energy Physics

Subleading bounds on chaos

Sandipan Kundu

Published Apr 4, 2022• 18 citations
arXivHTMLPDF

Abstract

Abstract Chaos, in quantum systems, can be diagnosed by certain out-of-time-order correlators (OTOCs) that obey the chaos bound of Maldacena, Shenker, and Stanford (MSS). We begin by deriving a dispersion relation for this class of OTOCs, implying that they must satisfy many more constraints beyond the MSS bound. Motivated by this observation, we perform a systematic analysis obtaining an infinite set of constraints on the OTOC. This infinite set includes the MSS bound as the leading constraint. In addition, it also contains subleading bounds that are highly constraining, especially when the MSS bound is saturated by the leading term. These new bounds, among other things, imply that the MSS bound cannot be exactly saturated over any duration of time, however short. Furthermore, we derive a sharp bound on the Lyapunov exponent λ2 ≤ $$ \frac{6\pi }{\beta } $$ 6 π β of the subleading correction to maximal chaos.

DOI
10.1007/jhep04(2022)010

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Bound on the exponential growth rate of out-of-time-ordered correlators

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It has been conjectured by Maldacena, Shenker, and Stanford [J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 08 (2016) 106] that the exponential growth rate of the out-of-time-ordered correlator (OTOC) $F(t)$ has a universal upper bound $2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$. Here we introduce a one-parameter family of out-of-time-ordered correlators ${F}_{\ensuremath{\gamma}}(t)$ ($0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$), which has as good properties as $F(t)$ as a regularization of the out-of-time-ordered part of the squared commutator $\ensuremath{\langle}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t),\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}(0)]}^{2}\ensuremath{\rangle}$ that diagnoses quantum many-body chaos, and coincides with $F(t)$ at $\ensuremath{\gamma}=1/2$. We rigorously prove that if ${F}_{\ensuremath{\gamma}}(t)$ shows a transient exponential growth for all $\ensuremath{\gamma}$ in $0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate $\ensuremath{\lambda}$ does not depend on the regularization parameter $\ensuremath{\gamma}$ and satisfies the inequality $\ensuremath{\lambda}\ensuremath{\le}2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$.

arXiv:1706.09160•Physical review. E•Jul 30, 2018•45 citations
Abstract Chaos, in quantum systems, can be diagnosed by certain out-of-time-order correlators (OTOCs) that obey the chaos bound of Maldacena, Shenker, and Stanford (MSS). We begin by deriving a dispersion relation for this class of OTOCs, implying that they must satisfy many more constraints beyond the MSS bound. Motivated by this observation, we perform a systematic analysis obtaining an infinite set of constraints on the OTOC. This infinite set includes the MSS bound as the leading constraint. In addition, it also contains subleading bounds that are highly constraining, especially when the MSS bound is saturated by the leading term. These new bounds, among other things, imply that the MSS bound cannot be exactly saturated over any duration of time, however short. Furthermore, we derive a sharp bound on the Lyapunov exponent λ2 ≤ 6πβ\frac{6\pi }{\beta }β6π​ 6 π β of the subleading correction to maximal chaos.
Abstract In maximally chaotic quantum systems, a class of out-of-time-order correlators (OTOCs) saturate the Maldacena-Shenker-Stanford (MSS) bound on chaos. Recently, it has been shown that the same OTOCs must also obey an infinite set of (subleading) constraints in any thermal quantum system with a large number of degrees of freedom. In this paper, we find a unique analytic extension of the maximally chaotic OTOC that saturates all the subleading chaos bounds which allow saturation. This extremally chaotic OTOC has the feature that information of the initial perturbation is recovered at very late times. Furthermore, we argue that the extremally chaotic OTOC provides a Källen-Lehmann-type representation for all OTOCs. This representation enables the identification of all analytic completions of maximal chaos as small deformations of extremal chaos in a precise way.
We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.
Abstract Out-of-time-order correlators (OTOCs) have emerged as the preeminent theoretical and experimental probe of quantum information scrambling — the delocalization of quantum information across many-body degrees of freedom. Here we present a comprehensive computational study of OTOCs across three paradigmatic models: the Sachdev-Ye-Kitaev (SYK) model, the non-integrable transverse-field Ising chain, and random unitary circuits. Through exact diagonalization and large-N analytics, we extract Lyapunov exponents, scrambling times, and butterfly velocities, demonstrating that all chaotic systems studied approach, but do not saturate, the Maldacena-Shenker-Stanford (MSS) bound λ_L ≤ 2π/β. Crucially, we introduce the Spectral Out-of-Time-Order Correlator (SOTOC), a novel frequency-domain diagnostic that decomposes information scrambling into its constituent spectral modes. The SOTOC achieves a 32% improvement in Lyapunov exponent extraction precision over conventional log-linear fitting and maintains robust performance at noise levels σ > 0.30, where traditional methods fail. Our phase diagram reveals a sharp transition in scrambling dynamics at h/J ≈ 1.0 in the TFIM, coinciding with the onset of quantum chaos as measured by level-spacing statistics. These results establish SOTOC as a practical and theoretically grounded tool for probing scrambling in near-term quantum devices and holographic systems.
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.
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.
It has been conjectured by Maldacena, Shenker, and Stanford [J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 08 (2016) 106] that the exponential growth rate of the out-of-time-ordered correlator (OTOC) F(t)F(t)F(t) has a universal upper bound $2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$. Here we introduce a one-parameter family of out-of-time-ordered correlators ${F}_{\ensuremath{\gamma}}(t)$ ($0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$), which has as good properties as F(t)F(t)F(t) as a regularization of the out-of-time-ordered part of the squared commutator $\ensuremath{\langle}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t),\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}(0)]}^{2}\ensuremath{\rangle}$ that diagnoses quantum many-body chaos, and coincides with F(t)F(t)F(t) at $\ensuremath{\gamma}=1/2$. We rigorously prove that if ${F}_{\ensuremath{\gamma}}(t)$ shows a transient exponential growth for all $\ensuremath{\gamma}$ in $0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate $\ensuremath{\lambda}$ does not depend on the regularization parameter $\ensuremath{\gamma}$ and satisfies the inequality $\ensuremath{\lambda}\ensuremath{\le}2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$.