High-frequency transport and zero-sound in an array of SYK quantum dots

Dear Editors,

We thank both referees for their reviews of our work. We took these reports into account while doing a major revision to our work. We have extended both the Introduction and Conclusions. We also simplified formulas for a more convenient presentation and added an Appendix for the readers interested in technical details. Below we answer the referees’ comments in detail.

Answers to the Report 1 on 2022-1-10

(i) Phase diagram (e.g. energy vs momentum) which demonstrates the hierarchy of energy/momentum scales and the expected window of the existence of the zero sound. We have added a phase diagram to describe regions with different features to the Conclusions (ii) Discussion of a usual hydrodynamic sound, its range of existence, and overlap (if any) with the zero-sound. The presented model lives on the lattice. The tunneling elements between dots do not depend on the lattice constants. Therefore, there is no traditional sound in the model. We have called the mode a zero-sound for the following reasons: - This mode describes the fluctuation of electron density - This mode exists only for high frequencies which are much higher than inverse thermalization time

(iii) Since the model is intrinsically disordered, one could expect a range where response functions exhibit diffusive behavior. Is it indeed the case and how does it conform to the presented expressions? The diffusive regime has detailed analysis in our paper (see section 5.3 of the new text, it was also present in the old version in section 5.2)

(iv) What are the relative contributions of reparametrization and phase fluctuations in the obtained response functions. E.g. how different are they in Majorana SYK (no phase fluctuations), vs. Fermi liquid (no reparametrizations)? We did not describe a crossover to Fermi liquid so our formulas do not cover this region of parameters. The comparison between the Fermi-liquid theory and our result is presented in the conclusion. We also mentioned in both versions of the text that in the absence of the charge transport there is only an intrinsic contribution to the heat conductivity.

(v) What is the role of finite N (if any) for the presented results? Finite N bounds the strength of \Gamma where our approach is valid. For \Gamma less than \Gamma_c ~ J/N, we can no longer consider fluctuations of the soft modes as Gaussian. The corresponding comment is added to the 1st paragraph of the Conclusions.

Answers to the Report 2 on 2022-2-4

1. A considerably improved introduction should be provided discussing the larger context of the model, the physical interpretation and consequences of zero-sound, etc. The conclusions should also be extended, since at this time they amount to a summary of results presenting no insight into their relevance, connections to other known systems, and so on. We have improved our Introduction and Conclusion.

2. At the end of section 3.2 the authors discuss the asymmetry parameter E, mentioning numerical evidence that its absolute value is found to be bounded. The provided citations, however, deal with a different variant of the complex SYK model, namely the mass-deformed case where the coefficient in front of the quadratic term in each quantum dot is not random but simply a fixed constant. This is different from the SYKq=4+SYKq=2 considered here, and can lead to confusion. To avoid this, the authors should clarify the distinctions between SYKq=4+SYKq=2 and the mass-deformed SYK model, and maybe comment on whether their results would also apply to a linear array of mass-deformed complex SYK quantum dots. In section 3.2 we are considering the saddle-point equations of the pure SYK model with real fermions. The authors of the mentioned papers considered different models but obtain the same saddle-point equations as ours. As a result, their numerical analysis of these equations is applicable to our case.

3. In equation (12) the authors introduce the parameter q because it "will be useful below for dimensional regularization of some singular expressions." Details of this regularization are scarce to non-existent, and the reader is left wondering whether the results obtained could depend on this regularization scheme. Some discussion of this procedure is therefore in order. We add the comment in the Appendix between (54) and (55)
4. While the manuscript is readable as is, the number of typographical errors can be distracting, and there is ample room for improving the English. Hopefully, the number of typographical errors and cases of improper use of English is now reduced considerably.

1. Introduction and conclusion were expanded.
2. Appendix was added.
3. The phase diagram with different transport regimes was added
4. The section " Noether’s theorem" was added
5. Formulas were simplified in the main part of the text
6. Large number of small correction was made