Semiclassical echo dynamics in the Sachdev-Ye-Kitaev model

TL;DR

This paper investigates semiclassical echo dynamics in the Sachdev-Ye-Kitaev model, demonstrating exponential divergence due to imperfections in time reversal, and identifies a Lyapunov exponent related to out-of-time-order correlators.

Contribution

It provides a semiclassical analysis of echo dynamics in the SYK model, linking exponential divergence to out-of-time-order correlators and identifying the Lyapunov exponent.

Findings

Exponential divergence in echo dynamics due to imperfections.

Lyapunov exponent is twice that of semiclassical equations.

Growth of out-of-time-order double commutator observed.

Abstract

The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows to identify a Lyapunov exponent $\lambda_L$. In particular, we find that $\lambda_L$ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator.