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

TL;DR

This work analyzes how out-of-time-ordered correlators (OTOCs) behave in single-body quantum systems and whether RPMD preserves the Maldacena bound across barrier scrambling. It shows that instantons and coherence dramatically influence OTOC growth, with RPMD sometimes violating the bound, while Matsubara dynamics reveal qualitatively different instanton behavior and more accurate scrambling rates. By investigating multidimensional barriers, asymmetric saddles, Boltzmann averaging, and scattering OTOCs, the study disentangles the roles of Boltzmann statistics, coherent interference, and topology of the instanton in quantum scrambling. The findings imply that quantum Boltzmann statistics alone do not guarantee the bound and motivate adopting Matsubara dynamics for reliable late-time OTOC predictions and potential insights into quantum reaction-rate theories, with practical implications for quantum chaos and information scrambling in few-body systems.

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.

Figures (11)

• Figure 1: Instanton formation for $T<T_{\rm c}$. $V(q)$ is the asymmetric Eckart model used in Ref.richardsonRingpolymerMolecularDynamics2009.
• Figure 2: A contour plot of $V(q)$ defined in Eq.\ref{['eqn:2dchaospotential']} for the values of $a=0.1$, $b=10$ used in simulation. Further details regarding simulation and convergence parameters can be found in Appendix \ref{['app:parametersforCounterExample']}.
• Figure 3: A counter-example to show the early-time exponential growth of RPMD $\lambda_{\rm rp}$ violating the chaos bound $\lambda_{\rm b}$. Simulation and convergence parameters can be found in Appendix \ref{['app:parametersforCounterExample']}.
• Figure 4: The unstable mode frequency $\eta_0$ does not satisfy the chaos bound for asymmetric systems for $T\leq T_{\rm c}$ [top panel]. This is due to significant projection of the unstable mode into non-centroid modes [bottom panel]. Both models taken from Ref.richardsonRingpolymerMolecularDynamics2009.
• Figure 5: Boltzmann averaging of the unstable mode for double well model of Ref.sadhasivamThermalQuenchingClassical2024. Empty circles show early time RPMD OTOC growth for the same system from sadhasivamThermalQuenchingClassical2024. Both results were reproduced with permission from this reference.