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Quantum echo dynamics in the Sherrington-Kirkpatrick model

by Silvia Pappalardi, Anatoli Polkovnikov, Alessandro Silva

This is not the current version.

Submission summary

As Contributors: Silvia Pappalardi · Anatoli Polkovnikov
Arxiv Link: (pdf)
Date submitted: 2020-02-17 01:00
Submitted by: Pappalardi, Silvia
Submitted to: SciPost Physics
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


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

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Quantum many-body systems
Current status:
Has been resubmitted

Author comments upon resubmission

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.

Reports on this Submission

Anonymous Report 1 on 2020-2-28 (Invited Report)


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.


1-OTOC considered is not the most widespread quantity


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

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Silvia Pappalardi  on 2020-02-17


We would like to correct two typos present in the Reply to the report I:

  • In question 1), the 8-th formula from the top of the reply reads:
\[ \left ( [ \hat a^{\dagger}(t), [\hat a^{\dagger}(t), \hat a^2(0)]\, ]\right )_w = 3 \left ( \frac{\partial\alpha(0)}{\partial \alpha(t)}\right )^2 + \left ( \frac{\partial\alpha^*(t)}{\partial \alpha^*(0)}\right )^2 + 3 \alpha(0)\frac{\partial^2 \alpha(0)}{\partial \alpha^2(t)} + \alpha^\ast(t)\frac {\partial^2 \alpha^{\ast}(t)}{\partial \alpha^{\ast\, 2}(0)} \ . \]
  • 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..."

Yours sincerly, Silvia Pappalardi, Anatoli Polkovnikov and Alessandro Silva