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Josephson current through the SYK model
by Luca Dell'Anna
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Submission summary
| Authors (as registered SciPost users): | Luca Dell'Anna |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202312_00054v2 (pdf) |
| Date submitted: | 2024-07-20 13:01 |
| Submitted by: | Dell'Anna, Luca |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We calculate the equilibrium Josephson current through a disordered interacting quantum dot described by a Sachdev-Ye-Kitaev model contacted by two BCS superconductors. We show that, at zero temperature and at the conformal limit, i.e. in the strong interacting limit, the Josephson current is suppressed by $U$, the strength of the interaction, as $\ln(U)/U$ and becomes universal, namely it gets independent on the superconducting pairing. At finite temperature $T$, instead, it depends on the ratio between the gap $\Delta$ and the temperature. A proximity effect exists but the self-energy corrections induced by the coupling with the superconducting leads seem subleading as compared to the self-energy due to the interaction for large number of particles. Finally we compare the results of the original four-fermion model with those obtained considering zero interaction, two-fermions and a generalized q-fermion model.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
hereafter I resubmit the paper to Scipost Physics.
In the last version I fixed some typos and included, at the end of page 9, the following sentence “…The same observation is valid if we want to improve the bare SYK solution including 1/N corrections. Also in that case the leading contribution to the Josephson current remains unchanged” when discussing the 1/N corrections questioned by Gnezdilov.
Actually, the derivation by Gnezdilov in his last report is in perfect agreement with my results while some 1/N corrections were already discussed in the text and Appendix.
I hope that now he will be happy with my previous reply and this addition.
About the comment of Bagrets, I should say that the request of taking the coupling matrix elements between the dot and leads as random variables was not present in his first report. Moreover, I would like to stress that the coupling proposed by Bagrets has to have vanishing mean value. If the mean value is finite his argument is no longer valid, while mine remains valid, with the inclusion of random fluctuations that, by the way, I discussed also in the paper.
Anyway studying another model implies making a totally new and different work. I can take his comment about considering a spinful SYK model as a good suggestion for further investigations. At this stage I will keep the model as it is.
My question is the following: given the model as in Eqs(1)-(4), are the calculations presented correct or not?
Since all the three Referees said that the topic presented in the paper is interesting and timely, since I replied to all the questions raised in the first round, included several additions so to double the volume of the paper following the suggestions of the Referees, and since none of the them, so far, confuted the correctness of my results, even after a long reviewing procedure, I resubmit the paper asking for further consideration.
Best regards,
Luca Dell'Anna
List of changes
At the end of page 9, I included the following sentence “…The same observation is valid if we want to improve the bare SYK solution including 1/N corrections. Also in that case the leading contribution to the Josephson current remains unchanged” .
Some typos have been fixed.
Current status:
Reports on this Submission
Report #2 by Dmitry Bagrets (Referee 1) on 2024-9-16 (Invited Report)
- Cite as: Dmitry Bagrets, Report on arXiv:scipost_202312_00054v2, delivered 2024-09-16, doi: 10.21468/SciPost.Report.9769
Strengths
The manuscripts attempts to solve an interesting and timely problem.
Weaknesses
1. Statistical properties of a coupling matrix between the SYK dot and superconducting leads are not analyzed properly.
2. The time-reversal broken version of the SYK Hamiltonan leads to zero Josephson current after disorder averaging.
Report
I think there is still some basic misunderstanding of my arguments. Calculations done by the author are formally correct until he gets the final result for the Josephson current, given by Eqs.~(58,59). It contains a sum over a multiple number of tunneling matrix elements, phases of which are not fixed. Therefore under very mild physical assumptions the overall sum will averaged to zero. And only mesoscopic sample-to-sample fluctuations of the Josephson current may remain. An assumption of statistical independence of such tunnel matrix elements is hidden at the core of mesoscopic physics. It doesn't mean that absolute values of tunneling couplings are zero, they are in fact not. I'm talking about pristine complex amplitudes. Without any additional symmetry constrains (which for example may enforce a reality condition to amplitudes), any sums involving such objects are zero, for example those formed by Eq. (15,16). The author's phrase after Eq. (21):
"Let us now make the reasonable assumption that tnσ = tmσ = tσ ... ", i.e. all amplitudes are real and equal to each other (!?)
contradicts to entire spirit of random SYK model. If all orbitals are already random insider a quantum dot (so that they generated random two-body integral J_ijkl), how dot-to-lead couplings could be expected to be deterministic?
It is also unfair to say that "the request of taking the coupling matrix elements between the dot and leads as random variables was not present in his first report". I can only cite my 1st report:
"The Ref. [2] can be also used as a proper guide how the random couplings between the quantum dot and leads can be incorporated into the model", where
[2] P. W. Brouwer and C. W. J. Beenakker, Chaos, Solitons & Fractals, Volume 8, Issues 7–8, July–August 1997, Pages 1249-1260, arXiv:cond-mat/9611162
To conclude I can't recommend a paper to the publication in its current form. And I can only repeat my offer to discuss more symmetry restrictive random ensemble discussed in
[1] Hanteng Wang, A. L. Chudnovskiy, Alexander Gorsky, and Alex Kamenev, Phys. Rev. Research 2, 033025,
where I may expect to see a non-zero Josephson current on average. Adjusting current calculations to such model is straightforward and doable. So, I disagree that it "...implies making a totally new and different work".
Requested changes
1. Reconsider the model of the SYK dot along the lines outlined in the my 2nd report and repeat calculations of the Josephson current.
Recommendation
Ask for major revision
Report #1 by Nikolay Gnezdilov (Referee 2) on 2024-8-25 (Invited Report)
- Cite as: Nikolay Gnezdilov, Report on arXiv:scipost_202312_00054v2, delivered 2024-08-25, doi: 10.21468/SciPost.Report.9651
Report
As I pointed out in my previous report, the terms ~ $O(1)$, e.g., due to $1/N$-corrections to the SYK saddle point ($1/N$-SYK), that the Author omitted, contribute in the same order to the free energy as the term stemming from the quantum dot-superconductor coupling (QD-SC). The key idea that the Author highlights in reply to my previous report is that only the phase-dependent part of the free energy matters for the Josephson effect. Indeed, one may drop the $1/N$-corrections to the SYK saddle point from the Josephson current computation. However, both contributions ($1/N$-SYK and QD-SC) appear in the same order in the free energy and the fermionic determinant. Yet, the way section 4 reads gives the impression that before evaluating the Josephson current, the Author consistently computes the determinant and, by extension, the free energy in the leading order, which is misleading and is very different from explicitly accounting for only the phase-dependent part of the free energy in the leading order and disregarding the rest. I believe this part of the paper requires major rewriting and clarification.
Recommendation
Ask for major revision