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

TL;DR

This paper investigates how instantons influence the growth of out-of-time correlation functions (OTOCs) in quantum systems, testing the effectiveness of RPMD and exploring the role of coherence and instantons in quantum chaos.

Contribution

It provides a detailed analysis of instantons' role in enforcing the Maldacena bound on OTOC growth and develops a new theoretical framework using Matsubara dynamics.

Findings

Instantons reduce OTOC growth rate, helping satisfy the Maldacena bound.

RPMD is not always sufficient to satisfy the bound, showing counterexamples.

Matsubara dynamics reveals different instanton behavior than RPMD, with stationary instantons in all but one coordinate.

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 $\Phi_0$, and fluctuations about it no longer resemble that of classical dynamics on a first order saddle as in RPMD.