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Euler-scale dynamical correlations in integrable systems with fluid motion

by Frederik S. Møller, Gabriele Perfetto, Benjamin Doyon, Jörg Schmiedmayer

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Submission summary

Authors (as registered SciPost users): Benjamin Doyon · Frederik Skovbo Møller · Gabriele Perfetto · Jörg Schmiedmayer
Submission information
Preprint Link:  (pdf)
Date submitted: 2020-11-05 11:59
Submitted by: Møller, Frederik Skovbo
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


We devise an iterative scheme for numerically calculating dynamical two-point correlation functions in integrable many-body systems, in the Eulerian scaling limit. Expressions for these were originally derived in Ref. [1] by combining the fluctuation-dissipation principle with generalized hydrodynamics. Crucially, the scheme is able to address non-stationary, inhomogeneous situations, when motion occurs at the Euler-scale of hydrodynamics. In such situations, in interacting systems, the simple correlations due to fluid modes propagating with the flow receive subtle corrections, which we test. Using our scheme, we study the spreading of correlations in several integrable models from inhomogeneous initial states. For the classical hard-rod model we compare our results with Monte-Carlo simulations and observe excellent agreement at long time-scales, thus providing the first demonstration of validity for the expressions derived in Ref. [1]. We also observe the onset of the Euler-scale limit for the dynamical correlations.

Current status:
Has been resubmitted

Author comments upon resubmission

Dear Editor,

Many thanks for your message and for forwarding us the report. We have answered all the questions of the referee report directly in the reply/comment below. We hope that the Referee will be satisfied by our revision and that the manuscript will be found ready for publication.

Best regards,
the Authors.

List of changes

We have added an additional sentence in the abstract concerning the subtlety of the indirect correlations.

The introduction has been expanded with a comprehensive discussion of the meaning of "thermalization" and "relaxation" in integrable systems. A discussion/definition of the Euler-scale has been added as well.

In the introduction, the subtle role of the indirect correlations is now further highlighted.

In the section 2 "Summary of GHD", we have moved the equations for the filling function and the dressing operation further up, whereby they now are eqs. (3) and (4).

In the section 2 "Summary of GHD", we have added a paragraph and eq. (7) detailing averages of local observables at the Euler-scale.

At the start of section 3 " Exact Euler-scale dynamical two-point correlations", we have added a definition of the Eulerian scaling limit for correlation functions.

In the section 3 " Exact Euler-scale dynamical two-point correlations", around eq. (18), we have made it more clear which terms correspond to the indirect correlations.

In section 4.2 "Bump release and Monte-Carlo comparison", we now further discuss the limits of the Euler-scale, and in which scenarios the GHD results do not apply.

We have added a new subsection 4.3 "Comparing light cones of different models" to the main text, which was previously found in the Appendix.

In Appendix C "Details of the Monte-Carlo simulations for the hard-rod gas", we have added an extension discussion of the limits of the Euler-scale results.

We have cited the references suggested by the referee in the manuscript.

Reports on this Submission

Anonymous Report 1 on 2020-11-11 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2007.00527v3, delivered 2020-11-11, doi: 10.21468/SciPost.Report.2180


The Authors have substantially improved the manuscript and, in my opinion, the revised version meets the acceptance criteria. However, I believe that the statement “integrable systems relax to stationary states which display non-thermal features, as a consequence of such conservation laws.” in Introduction is in the air with no mention of non-integrable systems which can either relax to the Gibbs state (the completely-chaotic systems with eigenstate thermalization) or to a non-thermal state (the incompletely-chaotic systems).

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Author:  Frederik Skovbo Møller  on 2020-11-17  [id 1047]

(in reply to Report 1 on 2020-11-11)

We thank the Referee for the second revision of the manuscript and for the supportive report. Indeed our exposition was not clear enough. We have revised the introduction of the manuscript by expanding the discussion about the relaxation of isolated many-body quantum systems at long times. We have specified more precisely the situation where the classification of stationary states is simplest: infinite-volume quenches from clustering states (which is the one we consider in this paper). There, one expects that the set of quasi-local conserved quantities fully specifies the stationary state; we do not think there are known exceptions to this. However, beyond this paradigm, indeed other possibilities occur,such as incompletely chaotic systems, as the Referee mentions. We have added a sentence pointing this out, including Ref. [24] about incompletely-chaotic systems, as suggested by the Referee. We have also included Refs. [21,22,23] about the eigenstate thermalization hypothesis (ETH) to make the discussion about the thermalization more complete. We emphasize, however, that we have discussed only some very fundamental points about thermalization of non-integrable systems, despite the rich literature available on this subject, since this topic is very marginally related to the original results presented in the manuscript.

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