Universal Chern number statistics in random matrix fields

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:

https://github.com/orswartzberg/Numerical-calculation-of-Chern-numbers

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.

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.