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Universal Chern number statistics in random matrix fields

by Or Swartzberg, Michael Wilkinson, Omri Gat

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Submission summary

Authors (as registered SciPost users): Michael Wilkinson
Submission information
Preprint Link:  (pdf)
Code repository:
Data repository:
Date accepted: 2023-05-17
Date submitted: 2023-03-31 11:27
Submitted by: Wilkinson, Michael
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approach: Theoretical


We investigate the probability distribution of Chern numbers (quantum Hall effect integers) for a parametric version of the GUE random matrix ensemble, which is a model for a chaotic or disordered system. The numerically-calculated single-band Chern number statistics agree well with predictions based on an earlier study [O. Gat and M. Wilkinson, SciPost Phys., 10, 149, (2021)] of the statistics of the quantum adiabatic curvature, when the parametric correlation length is small. However, contrary to an earlier conjecture, we find that the gap Chern numbers are correlated, and that correlation is weak but slowly-decaying. Also, the statistics of weighted sums of Chern numbers for many bands differs markedly from predictions based upon the hypothesis that gap Chern numbers are uncorrelated. All our results are consistent with the universality hypothesis described in the earlier paper, including in the previously unstudied regime of large correlation length, where the Chern statistics is highly non-Gaussian.

Author comments upon resubmission

In response to the referee reports, we have made two substantial revisions the text. Firstly, we have expanded the Introduction and Conclusions to include more discussion of the physical motivation and implications. Secondly, we have added an Appendix which explains the numerical methods, and which includes a link to a URL where the code is available.

Detailed responses to the two referee reports are appended below. We hope that the paper can now be accepted for publication.

Response to Anonymous Report 1

We thank the referee for their careful appraisal of our work.

The referee asks for more details of the numerical methods, and for the codes to be made publicly available. We have added an appendix, which explains the numerical approaches in some detail, and we have included a URL where the codes are available:

The referee also asks for more detail on the physical context. We have revised the Introduction accordingly. Our revision includes more general references about recent developments in topological approaches to condensed matter physics. We have also expanded upon the physical motivatioin for our investigation (in the Introduction) and upon the remaining unresolved issues (Conclusion).

Regarding the specific requested changes:

  1. The universal variance coefficient ~1.67 is expected to be applicable to complex quantum systems under a universality hypothesis, as discussed in the introduction. It is related to an integral of a correlation function in ref. [21], but we were not able to make an analytic estimate.

  2. Our attempts to explain the power-law relation for kurtosis were unsuccessful. Explaining this dead-end would be quite involved. We think there is nothing that we can usefully add to what is written.

  3. The text in section 6 states that the discrepancy between our numerical results and the prediction in ref. [21] is a result of correlations between the gap Chern numbers, whicg were assumed to be absent in the earlier work.

  4. The independence upon n is a consequence of the fact that random matrix models are ‘democratic’ in their treatment of individual states, and complex quantum systems should share this property. We have added a comment (below (21)) to that effect.

  5. As discussed above, we have added an appendix explaining the methods, and made the code available.

  6. We have expanded upon the un-resolved issues concerning the use of Chern numbers to describe the Hall effect in complex systems in the final paragraph of the Conclusions, indicating that we retain an ambition to address these in subsequent work.

Response to Anonymous Report 4

We thanks the referee for their generous comments and helpful criticisms.

We have expanded the Introduction to strengthen the physical motivation for the paper, as discussed in the response to referee 1.

We have also included an appendix giving a full description of the novel aspects of the numerical approach, and we have made the numerical codes available.

List of changes

We expanded discussion of physical motivation (Introduction) and of implications (Conclusion). An appendix which details the numerical approach has been added, including a link for obtaining the codes. A few minor issues in other sections were corrected.

Published as SciPost Phys. 15, 015 (2023)

Reports on this Submission

Report 1 by Daniel Arovas on 2023-5-1 (Invited Report)


I have nothing to add to my previous report. All my concerns have been appropriately responded to.


I have reviewed the resubmission and I find it acceptable for publication in SciPost. This paper has been delayed long enough.

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

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