# Quantum echo dynamics in the Sherrington-Kirkpatrick model

### Submission summary

 As Contributors: Silvia Pappalardi · Anatoli Polkovnikov Arxiv Link: https://arxiv.org/abs/1910.04769v3 (pdf) Date accepted: 2020-07-31 Date submitted: 2020-07-28 09:07 Submitted by: Pappalardi, Silvia Submitted to: SciPost Physics Discipline: Physics Subject area: Quantum Physics Approach: Theoretical

### 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.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### List of changes

- We added the definition of OTOC in the introduction as a "multi-point and multi-time correlation functions which cannot be represented on a single Keldysh contour".
- We have specified *chaotic* when referring to the semi-classical limit as a condition for the exponential growth.
- We have corrected the typos in the references [70-71].
- We implemented the corrections to the Bopp formalism.