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Quantum Scrambling and State Dependence of the Butterfly Velocity

by Xizhi Han, Sean A. Hartnoll

This Submission thread is now published as SciPost Phys. 7, 045 (2019)

Submission summary

As Contributors: Xizhi Han · Sean Hartnoll
Arxiv Link: (pdf)
Date accepted: 2019-09-20
Date submitted: 2019-08-15 02:00
Submitted by: Hartnoll, Sean
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


Operator growth in spatially local quantum many-body systems defines a scrambling velocity. We prove that this scrambling velocity bounds the state dependence of the out-of-time-ordered correlator in local lattice models. We verify this bound in simulations of the thermal mixed-field Ising spin chain. For scrambling operators, the butterfly velocity shows a crossover from a microscopic high temperature value to a distinct value at temperatures below the energy gap.

Ontology / Topics

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Quantum many-body systems

Published as SciPost Phys. 7, 045 (2019)

Author comments upon resubmission

We thank the referees for their constructive comments. We have made fairly extensive changes to the presentation of the paper to address the issues raised. We believe that the paper has been improved. The technical content is unchanged.

A. Both referees commented that we were using the words “quantum chaos” quite loosely. We agree with this comment. We have removed all references to chaos from the introduction and focused on notions of scrambling as defined by the OTOC.

B. Both referees commented that the conditions for vS = 0 were unclear, possibly undermining the usefulness of the scrambling velocity concept that we have introduced. This is a fair comment. In terms of the usefulness of our bound (on the temperature dependence of scrambling, in terms of vS), we mostly had in mind strongly scrambling systems. It is correct to say that for weakly scrambling systems, and even non-interacting systems, vS as currently defined may not be small, and therefore the bound is less useful. We have restructured the text to emphasize the strong scrambling case, and the connection to the numerics we performed, and to clarify the limitations of our current definition away from that limit (see for example the final paragraph on the section defining the scrambling velocity). We have also modified the discussion of the non-interacting model (discussion below fig 2) to make clearer why vS = 0 in that case.

Finally there were the following more minor comments from the referees:

C. One of the referees asked whether our results can be reformulated in terms of the energy of a pure state rather than the temperature. We do not have anything to contribute here beyond the usual expectation that the canonical and microcanonical ensembles should be equivalent for local observables in thermalizing systems.

D. One of the referees noted that "velocity dependent Lyapunov exponent" was used before it was defined. We have added a reference to the definition later in the text.

E. One of the referees asked whether there should be absolute value surrounding lambda on the RHS of equation (9), in order to obtain an upper bound in terms of vS. In general both terms on the RHS of equation (9) are important. We show in the text — around equation (23) — that in zooming in to the butterfly light cone, it is the first term that becomes important. This is why the bound is in terms of vS. The sign of lambda doesn’t matter here.

List of changes

See reply above.

Reports on this Submission

Anonymous Report 2 on 2019-9-16 (Invited Report)


I have previously submitted a report on this submission. The authors have addressed all my questions and I'm happy to recommend publication in its current form.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2019-9-4 (Invited Report)


I have previously commented on the timeliness and relevance of the paper.
The paper should be published in its current form.

  • validity: high
  • significance: good
  • originality: good
  • clarity: top
  • formatting: excellent
  • grammar: perfect

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Anonymous on 2019-09-04  [id 589]

I would like to thank the authors for addressing my questions.