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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_00054v3 (pdf) 
Date accepted:  20241009 
Date submitted:  20240925 11:39 
Submitted by:  Dell'Anna, Luca 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We calculate the equilibrium Josephson current through a disordered interacting quantum dot described by a SachdevYeKitaev model fully contacted by two BCS superconductors, such that all modes of the dot contribute to the coupling, which encodes hopping and spinflip processes. 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 gap. At finite temperature, instead, it depends on the ratio between the gap and the temperature. A proximity effect exists but the selfenergy corrections induced by the coupling with the superconducting leads seem subleading as compared to the selfenergy due to the interaction for large number of particles. Finally we compare the results of the original fourfermion model with those obtained considering zero interaction, twofermions and a generalized qfermion 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 multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Author comments upon resubmission
hereafter I resubmit the paper with other additions made in order to answer to hopefully the last question by Bagrets.
It is not clear to me the mechanism for getting fully randomized phases in the unknown wavefunctions appearing in the tunneling integrals, for a fully connected (alltoall) system where the particles are strongly correlated. The possibility of a complete decoherence in such a system seems to me a rare possibility. However, supposing that it comes from the arbitrariness of the phases, this argument should be true also for any multilevel/multichannel systems, even in the absence of disorder. In the previous version of the paper I discussed already the possibility of random fluctuations in the coupling, although assuming a residual coherence.
However, since I cannot exclude the possibility of such situation, where strong phase fluctuations occur, in the new version of the paper I extended the calculation for the Josephson current also in the presence of random phases in the tunneling amplitudes.
Contrary to what said by Bagrets, the presence of random phases, which can lead to vanishing mean values of the tunneling parameters, does not imply necessarily that also the averaged current (which depends on the products of them) should vanish. In the new version of the paper I showed it rigorously.
In the new additional Section 4.1.1, first I showed that the logarithmic form of the current appears already considering up to second order terms of the free energy, after expanding it in terms of the tunneling parameters. This result is valid since the strong interaction limit implies weak coupling, once the energy scale is fixed by U. Then in Section 4.1.2 I included orbitalspin dependent phases (which, then, can be chosen at will, also random) in the tunneling amplitudes. What I showed is that such dependence cancels out completely in the free energy, at least up to second order, and, therefore, in the current, reproducing the logarithmic form for the Josephson current, previously derived for a uniform phase.
In other words, the random phases in the tunneling amplitudes do not play any role in the Josephson current for the system under study and surely do not spoil it.
Finally, I would like also to remind that the paper contains also other results, which, in my opinion, warrant a publication, for instance the proof that the solvability of the SYK model is preserved also in the presence of a generic source of superconducting paring (see Sec. 3).
After answering rigorously to the last question of the Referee, showing by additional calculations that the result for the Josephson current is robust under phase fluctuations, I believe that now a quick positive decision can be made and the paper can be published without further unnecessary delay.
List of changes
 At the end of pag. 9, I included the following small paragraph to better clarify the issue raised by Gnezdilov: “Actually If we include those corrections in the bare Green’s function, the phase independent part of the free energy acquires a term of order O (1) but does not contribute to the Josephson current since it is phase independent, while the phase dependent part, which is O (1), and therefore, the Josephson current, acquires trivially a term O (1/N ), which is subleading and vanishes for large N . As a result, the Josephson current remains the same in the large N limit.”
 I added a couple of pages (from pag. 10 to pag. 12) to answer extensively to the last question raised by Bagrets: a new Section 4.1.1 titled “Perturbative analysis” and another new Section 4.1.2 titled “Random phases” where I showed that the Josephson current is not spoiled by random phases in the tunneling amplitudes.
Published as SciPost Phys. 17, 120 (2024)
Reports on this Submission
Report #1 by Dmitry Bagrets (Referee 1) on 2024102 (Invited Report)
Report
I suggest to accept the manuscript for the publication in its present form. My criticism from previous reports was honestly taken into consideration, and I now believe in the validity of obtained results.
Recommendation
Publish (meets expectations and criteria for this Journal)
Luca Dell'Anna on 20241002 [id 4822]
For the sake of clarity, in addition to what reported in the last version of the paper, we can easily generalize the calculation in the presence of generic random couplings, with also amplitude fluctuations, following the same steps done for the random phases. One can show that, at least up to second order of the tunneling parameters, the free energy is the same as that obtained for uniform couplings, providing that the modulus of the uniform tunneling amplitude is taken as the square root of the average of the squared tunneling amplitudes (see the version posted in the arxiv for details, pag. 13). As already stated in the version posted here for the random phases, the Josephson current is far from being zero as the Referee was afraid of. This proof, as well as that shown already in this version, gives a definitive answer to the last question raised by the Referee.