Quantum echo dynamics in the Sherrington-Kirkpatrick model

TL;DR

This paper investigates quantum echo dynamics in the Sherrington-Kirkpatrick model, revealing conditions for exponential divergence of observables due to chaos, and compares semiclassical and quantum behaviors.

Contribution

It demonstrates how chaos induces exponential echo growth in the SK model under specific conditions, highlighting the role of collective observables and semiclassical limits.

Findings

Echo grows exponentially up to Ehrenfest time in the SK model.

Ehrenfest time scales logarithmically with system size N.

Short-range SK model shows polynomial echo growth, independent of N.

Abstract

Understanding the footprints of chaos in quantum-many-body systems has been under debate for a long time. In this work, we study the echo dynamics of the Sherrington-Kirkpatrick (SK) model with transverse field under effective time reversal. We investigate numerically its quantum and semiclassical dynamics. We explore how chaotic many-body quantum physics can lead to exponential divergence of the echo of observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state and iii) the existence of a well-defined chaotic semi-classical (large-$N$) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins $N$. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a short-range version of the SK model, where the Ehrenfest time does not depend on $N$ and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe it experimentally.