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arXiv:2512.00471

Out-of-time correlation functions in single-body systems

Andrew C. Hunt

quant-phphysics.chem-phSubmitted Nov 29, 2025• Announced Dec 2, 2025• 0 citations
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Abstract

In the study of quantum chaos, `out of time ordered correlators' (OTOCs) are commonly used to quantify the rate at which quantum information is scrambled. This rate has been conjectured by Maldecena et al. to obey a universal, temperature dependent bound. Recent studies have shown that instantons, delocalised structures that dominate tunnelling statistics over barriers, reduce the growth rate of OTOCs. For the case of the symmetric double well, this reduction ensures the bound is maintained for OTOCs generated using ring polymer molecular dynamics (RPMD), a method with approximate dynamics but exact quantum statistics. In this report we set out to further understand the role of the instanton in the enforcement of the Maldacena bound and test whether RPMD is sufficient to satisfy the bound. We also investigate the impact of coherence on the flattening of of OTOCs by contrasting bounded with scattering systems. For the scattering system we observe a significantly smaller OTOC growth rate than that of the analogous bounded system, and a flattening in growth rate as time progresses. We attribute the first effect to influence of the Boltzmann operator, and the second to interference caused by anharmonicity of the potential. In our studies of RPMD, we find counterexamples showing that it is not sufficient to satisfy the bound. We develop a theory for OTOCs using (analytically-continued) Matsubara dynamics, revealing significantly different dynamical behaviour around the instanton compared to the predictions of RPMD. The instanton is found to be stationary in all coordinates but its collective angle $Φ_0$, and fluctuations about it no longer resemble that of classical dynamics on a first order saddle as in RPMD.

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Out-of-time-order correlators (OTOCs) can be used to probe how quickly a quantum system scrambles information when the initial conditions of the dynamics are changed. In sufficiently large quantum systems, one can extract from the OTOC the quantum analog of the Lyapunov coefficient that describes the timescale on which a classical chaotic system becomes scrambled. OTOCs have been applied only to a very limited number of toy models, such as the Sachdev-Ye-Kitaev model connected with black hole information scrambling, but they could find much wider applicability for information scrambling in quantum systems that allow comparison with experiments. The vibrations of polyatomic molecules are known to undergo a transition from regular dynamics at low energy to facile energy flow at sufficiently high energy. Molecules therefore represent ideal quantum systems to study scrambling in many-body systems of moderate size (here 6 to 36 degrees of freedoms). By computing quantum OTOCs and their classical counterparts we quantify how information becomes ``scrambled'' quantum mechanically in molecular systems. Between early ``ballistic'' dynamics, and late ``saturation'' of the OTOC when the full density of states is explored, there is indeed a regime where a quantum Lyapunov coefficient can be defined for all molecules in this study. Comparison with experimental rate data shows that slow scrambling as measured by the OTOC can reach the timescale of molecular reaction dynamics. Even for the smallest molecules we discuss, the Maldacena bound remains satisfied by regularized OTOCs, but not by unregularized OTOCs, highlighting that the former are more useful for discussing information scrambling in this type of moderate-size quantum system.

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Universal Logarithmic Scrambling in Many Body Localization

Yu Chen

Out of time ordered correlator (OTOC) is recently introduced as a powerful diagnose for quantum chaos. To go beyond, here we present an analytical solution of OTOC for a non-chaotic many body localized (MBL) system, showing distinct feature from quantum chaos and Anderson localization (AL). The OTOC is found to fall only if the nearest distance between the two operators being shorter than $ξ\ln t$, where $ξ$ is dimensionless localization length. Thereafter, we found an universal power law decay of OTOC as $2^{-ξ\ln t}$, implying an universal logarithmic growth of second Rényi entropy, where $ξ$ plays the role of information scrambling rate. A relation between butterfly velocity and scrambling rate is found.

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In the study of quantum chaos, `out of time ordered correlators' (OTOCs) are commonly used to quantify the rate at which quantum information is scrambled. This rate has been conjectured by Maldecena et al. to obey a universal, temperature dependent bound. Recent studies have shown that instantons, delocalised structures that dominate tunnelling statistics over barriers, reduce the growth rate of OTOCs. For the case of the symmetric double well, this reduction ensures the bound is maintained for OTOCs generated using ring polymer molecular dynamics (RPMD), a method with approximate dynamics but exact quantum statistics. In this report we set out to further understand the role of the instanton in the enforcement of the Maldacena bound and test whether RPMD is sufficient to satisfy the bound. We also investigate the impact of coherence on the flattening of of OTOCs by contrasting bounded with scattering systems. For the scattering system we observe a significantly smaller OTOC growth rate than that of the analogous bounded system, and a flattening in growth rate as time progresses. We attribute the first effect to influence of the Boltzmann operator, and the second to interference caused by anharmonicity of the potential. In our studies of RPMD, we find counterexamples showing that it is not sufficient to satisfy the bound. We develop a theory for OTOCs using (analytically-continued) Matsubara dynamics, revealing significantly different dynamical behaviour around the instanton compared to the predictions of RPMD. The instanton is found to be stationary in all coordinates but its collective angle Φ0Φ_0Φ0​, and fluctuations about it no longer resemble that of classical dynamics on a first order saddle as in RPMD.
Out-of-time-order correlators (OTOCs) can be used to probe how quickly a quantum system scrambles information when the initial conditions of the dynamics are changed. In sufficiently large quantum systems, one can extract from the OTOC the quantum analog of the Lyapunov coefficient that describes the timescale on which a classical chaotic system becomes scrambled. OTOCs have been applied only to a very limited number of toy models, such as the Sachdev-Ye-Kitaev model connected with black hole information scrambling, but they could find much wider applicability for information scrambling in quantum systems that allow comparison with experiments. The vibrations of polyatomic molecules are known to undergo a transition from regular dynamics at low energy to facile energy flow at sufficiently high energy. Molecules therefore represent ideal quantum systems to study scrambling in many-body systems of moderate size (here 6 to 36 degrees of freedoms). By computing quantum OTOCs and their classical counterparts we quantify how information becomes ``scrambled'' quantum mechanically in molecular systems. Between early ``ballistic'' dynamics, and late ``saturation'' of the OTOC when the full density of states is explored, there is indeed a regime where a quantum Lyapunov coefficient can be defined for all molecules in this study. Comparison with experimental rate data shows that slow scrambling as measured by the OTOC can reach the timescale of molecular reaction dynamics. Even for the smallest molecules we discuss, the Maldacena bound remains satisfied by regularized OTOCs, but not by unregularized OTOCs, highlighting that the former are more useful for discussing information scrambling in this type of moderate-size quantum system.
The rate at which information scrambles in a quantum system can be quantified using out-of-time-ordered correlators. A remarkable prediction is that the associated Lyapunov exponent [Formula: see text] that quantifies the scrambling rate in chaotic systems obeys a universal bound [Formula: see text]. Previous numerical and analytical studies have indicated that this bound has a quantum-statistical origin. Here, we use path-integral techniques to show that a minimal theory to reproduce this bound involves adding contributions from quantum thermal fluctuations (describing quantum tunneling and zero-point energy) to classical dynamics. By propagating a model quantum-Boltzmann-conserving classical dynamics for a system with a barrier, we show that the bound is controlled by the stability of thermal fluctuations around the barrier instanton (a delocalized structure which dominates the tunneling statistics). This stability requirement appears to be general, implying that there is a close relation between the formation of instantons, or related delocalized structures, and the imposition of the quantum-chaos bound.
The rate at which information scrambles in a quantum system can be quantified using out-of-time-ordered correlators. A remarkable prediction is that the associated Lyapunov exponent λ that quantifies the scrambling rate in chaotic systems obeys a universal bound λ < 2πkBT/h. Previous numerical and analytical studies have indicated that this bound has a quantum-statistical origin. Here, we use path-integral techniques to show that a minimal theory to reproduce this bound involves adding contributions from quantum thermal fluctuations (describing quantum tunneling and zero-point energy) to classical dynamics. By propagating a model quantum-Boltzmannconserving classical dynamics for a system with a barrier, we show that the bound is controlled by the stability of thermal fluctuations around the barrier instanton (a delocalized structure which dominates the tunneling statistics). This stability requirement appears to be general, implying that there is a close relation between the formation of instantons, or related delocalized structures, and the imposition of the quantum-chaos bound.
The ultimate regularity of quantum mechanics creates a tension with the assumption of classical chaos used in many of our pictures of chemical reaction dynamics. Out-of-time-order correlators (OTOCs) provide a quantum analog to the Lyapunov exponents that characterize classical chaotic motion. Maldacena, Shenker, and Stanford have suggested a fundamental quantum bound for the rate of information scrambling, which resembles a limit suggested by Herzfeld for chemical reaction rates. Here, we use OTOCs to study model reactions based on a double-well reaction coordinate coupled to anharmonic oscillators or to a continuum oscillator bath. Upon cooling, as one enters the tunneling regime where the reaction rate does not strongly depend on temperature, the quantum Lyapunov exponent can approach the scrambling bound and the effective reaction rate obtained from a population correlation function can approach the Herzfeld limit on reaction rates: Tunneling increases scrambling by expanding the state space available to the system. The coupling of a dissipative continuum bath to the reaction coordinate reduces the scrambling rate obtained from the early-time OTOC, thus making the scrambling bound harder to reach, in the same way that friction is known to lower the temperature at which thermally activated barrier crossing goes over to the low-temperature activationless tunneling regime. Thus, chemical reactions entering the tunneling regime can be information scramblers as powerful as the black holes to which the quantum Lyapunov exponent bound has usually been applied.
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 $\ensuremath{\lambda}\ensuremath{\le}2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$.
Out of time ordered correlator (OTOC) is recently introduced as a powerful diagnose for quantum chaos. To go beyond, here we present an analytical solution of OTOC for a non-chaotic many body localized (MBL) system, showing distinct feature from quantum chaos and Anderson localization (AL). The OTOC is found to fall only if the nearest distance between the two operators being shorter than ξln⁡tξ\ln tξlnt, where ξξξ is dimensionless localization length. Thereafter, we found an universal power law decay of OTOC as 2−ξln⁡t2^{-ξ\ln t}2−ξlnt, implying an universal logarithmic growth of second Rényi entropy, where ξξξ plays the role of information scrambling rate. A relation between butterfly velocity and scrambling rate is found.