Detecting few-body quantum chaos: out-of-time ordered correlators at saturation

TL;DR

This paper investigates the behavior and saturation of out-of-time ordered correlators in few-body quantum systems, revealing universal temperature-dependent saturation values that serve as indicators of quantum chaos.

Contribution

It provides a detailed analysis of OTOC saturation in few-body systems, showing universal temperature dependence and challenging the notion of exponential growth in such systems.

Findings

Saturation value of OTOC decreases with temperature in a universal manner.

No clear exponential growth regime in the studied few-body systems.

Saturated OTOC can serve as an indicator of quantum chaos.

Abstract

We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the plateau (saturated) value of OTOC reached at long times decreases with temperature in a simple and universal way: $\exp(\mathrm{const.}/T^2)$ for strong chaos (including random matrices) and $\exp(\mathrm{const.}/T)$ for weak chaos. For small matrices and sufficiently complex operators, there is also another, high-temperature regime where the saturated OTOC grows with temperature. Therefore, the plateau OTOC value is a meaningful indicator of few-body quantum chaos. We also discuss some general consequences of our findings for the AdS/CFT duality.