# Quantum echo dynamics in the Sherrington-Kirkpatrick model

### Submission summary

 As Contributors: Silvia Pappalardi · Anatoli Polkovnikov Arxiv Link: https://arxiv.org/abs/1910.04769v2 (pdf) Date submitted: 2020-02-17 01:00 Submitted by: Pappalardi, Silvia Submitted to: SciPost Physics Academic field: Physics Specialties: 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, by investigating 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 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.

### Ontology / Topics

See full Ontology or Topics database.

###### Current status:
Has been resubmitted

### List of changes

- We re-wrote the introduction in a more cohesive manner.
- We added a small comparison between the SK and SYK models.
- We added a new appendix containing the derivation of the semi-classical limit of the echo observable and the square commutator.
- We added a new appendix containing the derivation of TWA and its validity for the SK model.
- We re-wrote the introduction to TWA in Section 5.
- We changed the caption of the pictures and added ordinate labels when missing. In Fig.4, we added an exponential function to guide the reader's eyes to the thermodynamic limit.
- We improved the discussion of the numerical findings in Section 6.
- Updated bibliography.

### Submission & Refereeing History

Resubmission 1910.04769v3 on 28 July 2020

Resubmission 1910.04769v2 on 17 February 2020
Submission 1910.04769v1 on 15 October 2019

## Reports on this Submission

### Strengths

1-timely subject (OTOCs)
2-cutting edge analytics and semiclassics
3-Relevant results about time-scales and possibility to observe exponetial growth of the OTOC.

### Weaknesses

1-OTOC considered is not the most widespread quantity

### Report

The authors have answered the concerns of the referees rather carefully and I think the manuscript in the present form is a significant contribution to the subject of characterizing chaos in complex quantum systems. I recommend publication.

• validity: high
• significance: good
• originality: top
• clarity: ok
• formatting: good
• grammar: good

$\left ( [ \hat a^{\dagger}(t), [\hat a^{\dagger}(t), \hat a^2(0)]\, ]\right )_w = 3 \left ( \frac{\partial\alpha(0)}{\partial \alpha(t)}\right )^2 + \left ( \frac{\partial\alpha^*(t)}{\partial \alpha^*(0)}\right )^2 + 3 \alpha(0)\frac{\partial^2 \alpha(0)}{\partial \alpha^2(t)} + \alpha^\ast(t)\frac {\partial^2 \alpha^{\ast}(t)}{\partial \alpha^{\ast\, 2}(0)} \ .$
• In question 4), the third point to the reply to the minor remarks reads: "We thank the referee for pointing out a mistake in the notations. Generally, if one assumes that the time-evolution of a quantum state with an hamiltonian $\hat H$ is $e^{-i \hat H t}|\psi_0\rangle$, then..."