Josephson current through the SYK model

This is an interesting question to address theoretically.

1. Model is not justified by realistic experimental setup.
2. Author refers at many places to numerical calculations, but did not
provide any plots in support of accuracy of such calculations.
3. The references are largely incomplete

I find the question addressed in this work interesting.
However, I would like to get a clarification on the following questions:

* Is the replica symmetric (diagonal) solution justified?
(See Hanteng Wang, D. Bagrets, A. L. Chudnovskiy, A. Kamenev . On the replica structure of Sachdev-Ye-Kitaev model. J. High Energy Phys. 9 (2019) 57 )

* How does the "Negative-U" Hubbard term influence the solution and main conclusions of the paper? (See
Hanteng Wang, A. L. Chudnovskiy, Alexander Gorsky, Alex Kamenev.
SYK Superconductivity: Quantum Kuramoto and Generalized Richardson Models. Phys. Rev. Research 2, 033025 (2020) )

* Does the phonon bath affect the main results and conclusions? (See e.g. Hossein Hosseinabadi, Shane P. Kelly, Jörg Schmalian, Jamir Marino. Thermalization of non-Fermi-liquid electron-phonon systems: Hydrodynamic relaxation of the Yukawa-Sachdev-Ye-Kitaev model. Phys. Rev. B 108, 104319 (2023) and many references therein).

1. Get clarification on the listed items and modify the model if necessary.

2. Visualize the claims about numerical solutions and provide quantified justification of some approximations used in the paper.

3. Update the reference list to include some relevant publications on the superconducting SYK models

1. It is a well-stated problem with an outcome of interest to characterize the SYK model realized in a solid-state platform.

1. The derivation of the main result is not transparent (See p1-p3 in the report).
2. Absence of the comparison with the known results (See p4 in the report).

The paper addresses the Josephson effect in the disordered quantum dot described by the Sachdev-Ye-Kitaev (SYK) model sandwiched between the two bulk superconductors. The primary result of the paper is the independence of the current's amplitude of the order parameter in the superconductors. The problem is well stated and may be especially interesting in light of recent attention to realizing the SYK model in solid-sate or atomic platforms. Yet, the derivation and discussion of the Josephson current in the paper raises several questions specified below, which make me hesitant to recommend the manuscript for publication at this stage.

1. In the paper, the decomposition of the second term in the effective action of the quantum dot (16) in Section 3 gives rise to terms proportional to $N^3$ in Eq. (17). This turns out to be essential for neglecting some terms in the quantum dot's self-energy in the analysis below. However, a conventional for the SYK-like models decomposition of the same term using Lagrangian multipliers in a spin-singlet channel leads to the contributions only proportional to $N$. Is it possible to clarify the origin of $N^3$ terms?

2. Section 4 starts by highlighting the negligibility of the coupling to the superconductors for the dot's self-energy in the large-$N$ limit. However, in the following line, Eq. (41) still contains $1/N$ contribution to the dot's self-energy via ${\cal T}$, arising from integrating bulk superconductors at the stage of Eq. (16). This finite $N$ contribution seems crucial for computing the Josephson current in Eq. (48).

3. The Josephson current through the disordered quantum dot may vary from sample to sample [1]. When computing the current via the phase derivative of the self-energy in Eq. (48), the Author uses the replica-diagonal solution of the Schwinger-Dyson equations to evaluate the partition function. This assumption implies an equivalence between $\overline{\log Z}$ and $\log {\overline Z}$, where $\overline{\ldots}$ denotes the disorder average and $Z$ is the partition function [2]. In the SYK model, the disorder average is usually done at the level of the partition function if one computes the observables in the large-$N$ limit. However, since the computation of the Josephson current requires keeping the finite $N$ corrections, the analysis of the replica off-diagonal solutions that may lead to sample-to-sample fluctuations of the Josephson current has to be included.

4. The primary result of the paper is Eq. (56), which outlines the independence of the Josephson current through the SYK quantum dot from the order parameter in the superconductors. How does it compare to the known results on the Josephson effect in disordered quantum dots obtained with the Random Matrix Theory [3]? Is it possible to recover any of that using the same approach but replacing the SYK quantum dot with the random free-fermion model?

5. The SYK Hamiltonian (4) describing the quantum dot is spin-polarized since it does not possess a spin degree of freedom. Then, the quantum dot acts as magnetic impurities, which may suppress superconductivity at low temperatures and, hence, the Josephson effect. For the SYK case, see Ref. [4].

6. In the Introduction, the Author overviews some literature on transport in the SYK-like systems. Yet, there is no literature on superconductivity in the SYK model and the Josephson effect in disordered quantum dots that is seemingly most relevant for the study. Including some references certainly would benefit the reader.

References:

[1] C. W. J. Beenakker, Three “universal” mesoscopic Josephson
effects, Transport Phenomena in Mesoscopic Systems, edited by H. Fukuyama and T. Ando (Springer, Berlin, 1992).
[2] A. Kitaev and S. J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, J. High Energ. Phys. 2018, 183 (2018).
[3] P. W. Brouwer and C. W. J. Beenakker, Anomalous temperature dependence of the supercurrent through a chaotic Josephson junction, Chaos, Solitons & Fractals 8, 1249 (1997).
[4] Y. Cheipesh, A. I. Pavlov, V. Scopelliti, J. Tworzydło, and N. V. Gnezdilov, Reentrant superconductivity in a quantum dot coupled to a Sachdev-Ye-Kitaev metal, Phys. Rev. B 100, 220506(R) (2019).

1. The derivation of the main result requires clarification.
2. A distinctive comparison to the known results is needed.

The manuscripts attempts to solve an interesting and timely problem.

1. The micrscopic model violates basic rules of describing the superconding systems. See more details in my report below.

2. Statistical properties of a coupling matrix between the SYK dot and superconducting leads are not analyzed properly.

The manuscript by Luca Dell'Anna contains an attempt to study an interesting question of a possibility the Josephson current to flow through the interacting SYK quantum dot. I have some strong doubts about the affirmative answer to this question, which is given by the author. Therefore, before it is clarified, I can't recommend the paper to the publication.

My main concern is related to the basic symmetries of the model and the form of microscopic Hamiltonians (3,4) which describe an electron tunneling between the quantum dot and the leads and the SYK interaction, respectively. Judging from the leads' Hamiltonian (1), I see that the latter describes SU(2) time-reversal symmetric superconductor. That's why its description can be reduced to two-component Nambu spinors of the form (5). However, spin structure is entirely ignored in the interaction Hamiltonian (4) and, correspondingly, in the tunneling term (3). This is very unsatisfactory. For instance, time-reversal and SU(2) symmetric form of the SYK interaction contains two pieces, as one can check, for instance in Ref [1]. Other variants of the complex SYK model, violating either SU(2) or time-reversal symmetry are potentially possible. The choice of a symmetry class should also affect the statistical properties of tunneling matrix elements in (3), which might become complex or spin-dependent.

I suggest the author corrects this principal mistake by restoring the spin structure of fermions on the SYK dot. Depending on the presence or absence of the time-reversal symmetry I may only guess that the Josephson effect might be present or entirely suppressed in this model. This needs to be clarified by rigorous calculations.

My another comment refers to Sec. 4.3 "Zero interaction limit". Here it makes more sense to compare the new results to Ref. [2]. Similar to (yet to be verified) author's answer (56), the Josephson current via non-interacting chaotic quantum dot contains the logarithmic dependence on the superconducting phase, see Eq. (31a) in [2]. Also, the dependence on the Thouless energy scale (which is equal to \Gamma in the present paper) happens to be identical. Such universality should be emphasized (and possibly explained qualitatively). 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. It may superimpose a 'naive' assumption of the author made after Eq. (14), namely, that matrix elements are all identical independent of their quantum numbers.

[1] Hanteng Wang, A. L. Chudnovskiy, Alexander Gorsky, and Alex Kamenev, Phys. Rev. Research 2, 033025
[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

1. Change the model according to basic principles and reconsider all subsequent calculations and conclusions.