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Scale invariant distribution functions in quantum systems with few degrees of freedom

by Emanuele G. Dalla Torre

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Emanuele Dalla Torre
Submission information
Preprint Link:  (pdf)
Date accepted: 2018-08-23
Date submitted: 2018-07-19 02:00
Submitted by: Dalla Torre, Emanuele
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


Scale invariance usually occurs in extended systems where correlation functions decay algebraically in space and/or time. Here we introduce a new type of scale invariance, occurring in the distribution functions of physical observables. At equilibrium these functions decay over a typical scale set by the temperature, but they can become scale invariant in a sudden quantum quench. We exemplify this effect through the analysis of linear and non-linear quantum oscillators. We find that their distribution functions generically diverge logarithmically close to the stable points of the classical dynamics. Our study opens the possibility to address integrability and its breaking in distribution functions, with immediate applications to matter-wave interferometers.

Published as SciPost Phys. 5, 023 (2018)

Author comments upon resubmission

Dear Editor,
I sincerely thank all 3 Referees for carefully reading my manuscript and for their useful comments.
The Referees asked minor changes to the manuscript, which I implemented in the new version.
I think that the article should now be ready for publications.
Thank you for your time and consideration.
Best regards,
Emanuele Dalla Torre

List of changes

- Added a paragraph to the introduction, were I define the concept of scale invariance and explain the difference between the usual case and the new one.
- Added two appendices on the effects of the finite size (A.3) and on the scaling of additional observables (A.4).
- Added two footnotes on the formal definition of scale invariance (page 2) and Gaussian fixed points (page 3).
- Improved the discussion on the effects of non-linearities (page 3), finite size (page 5), and random variables (page 7).

Reports on this Submission

Anonymous Report 3 on 2018-8-17 (Invited Report)


I have read the new version of the manuscript by Dalla Torre entitled \emph{Scale invariant distribution functions in quantum systems with few degrees of freedom}, which has been resubmitted to SciPost.

The author has significantly changed the manuscript according to the referees comments. In particular the new version has now a more extended introduction and the appendix contains information on the dependence from the total spin S and the study of distribution function for different observables. The author has also answered convincingly to several issues raised by the Referees.

Overall, I think the manuscript contains interesting new results and deserves in this form publication in SciPost.

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

Anonymous Report 2 on 2018-8-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1709.01942v4, delivered 2018-08-14, doi: 10.21468/SciPost.Report.554


see previous report


see previous report


I read through the author’s reply and through the relevant parts in the manuscript, and I can be brief, my comments are comprehensively addressed.
The notion of scale invariance put forward by the author is unconventional, only approximate, and one may have referenced what he means differently, but it is now clearly defined and no one will get confused based on that. So I would leave the responsibility for such choice to the author.

Requested changes

optional, see above

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

Anonymous Report 1 on 2018-7-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1709.01942v4, delivered 2018-07-28, doi: 10.21468/SciPost.Report.541


see previous report


see previous report and minor amendments


The author amended his ms. according to the requests by the referees. Concerning his answers to my questions:

“1. The first comment by the Referee deals with the core novelty of this work. In the usual case, scale invariance is defined with respect to a rescaling of the space and time coordinates. In contrast, here I consider the rescaling of physical observables. This point is not better explained in the 3rd paragraph of the Introduction [“Our definition…”]. My scale invariance does not reveal itself as a universal correlation function, but rather as a universal distribution function.”

I do not think that there is a fundamental difference between the scaling considered by the author and conventional scaling: If one reduces the dimension to 0+1 and averages over time then the remaining space is zero-dimensional and scaling “only” affects observables. This is the same in a higher-dimensional system where operators also receive a scaling dimension which summarizes their scaling resulting from a rescaling of time (and spatial) arguments.

Also scaling of distribution functions as compared to correlation functions does not represent a difference as the distribution can be expanded and thus represented and generated by its moments which are the correlation functions. Hence, if all correlation functions scale, also the distribution function scales. The nature of the problems considered by the author suggests to consider distribution functions in the first place.

This is why I asked for the relation between the scaling of distributions and that of corr. functions in the paper. Maybe one could at least add a remark on that.

“The Referee correctly points… invariant tori.””

I agree.

I am also happy with answer and amendment concerning point 2.

Requested changes

Optional: see report

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

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