Refined quantum Lyapunov exponents from replica out-of-time-order correlators

TL;DR

This paper introduces a new quantum chaos indicator based on a logarithmic out-of-time-order correlator, linking quantum and classical chaos, and provides analytical calculation methods with applications to various models.

Contribution

It proposes a novel quantum chaos measure derived from replica out-of-time-order correlators, connecting quantum and classical chaos, and offers analytical tools for its computation.

Findings

The new indicator reproduces classical Lyapunov exponents in the semiclassical limit.

Replica correlations can decrease the estimated Lyapunov exponent.

Application to models like the quantum cat map and SYK demonstrates the method's effectiveness.

Abstract

We suggest a new indicator of quantum chaos based on the logarithmic out-of-time-order correlator. On the one hand, this indicator correctly reproduces the average classical Lyapunov exponent in the semiclassical limit and directly links the definitions of quantum chaos and classical K-system. On the other hand, it can be analytically calculated using the replica trick and the Schwinger-Keldysh diagram technique on a $2n$-fold Keldysh contour. To illustrate this approach, we consider several one-dimensional systems, including the quantum cat map, and three paradigmatic large-$N$ models, including the Sachdev-Ye-Kitaev model. Furthermore, we find that correlations between replicas can reduce the magnitude of the Lyapunov exponent compared to estimates based on conventional out-of-time-order correlators.