Impurity Knight shift in quantum dot Josephson junctions

1-Significantly extends a new direction in the research field of superconducting quantum dots, namely the Knight shift and spin response to microwave drives
2-Systematic wide scope study of various aspects of the problem including massive NRG calculations, perturbation theory as well as relation to the experiments

1-Assembly of many interesting and relevant aspects of the studied model, which, however, are not too well bound into a single coherent story
2-Presentation of results especially in the perturbation theory part of the manuscript
3-Several confusing inconsistencies throughout the text

The paper studies the Knight shift in a quantum dot coupled to two phase-biased superconducting leads and thus forming a nanoscopic Josephson junction. This topic is fresh, motivated by a recent experiment and the paper thus satisfies one of the necessary criteria of acceptance "Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work;" and deserves publication in SciPost Physics.

On the other hand, its current form is not quite suitable for direct publication and should be further improved as I will detail below. Apart from a number of minor suggestions I see two global shortcomings.
First, I was really disappointed when finally coming to Sec. VII about the experimental evidence. The data were analyzed to show the existence of a term linear in the magnetic field and proportional to the cos of the phase difference in line with the Knight shift assumption but not even an attempt to apply the theory developed on the previous 17 pages had been made. In this respect the experimental part should be taken solely as a motivation of the otherwise purely theoretical study, i.e., this is not in my view a joint theory-experiment paper, which was my impression when reading the abstract.
Second, it's not very clear to me why the authors spent so much effort on the perturbation theory since its connection to the nonperturbative NRG results is not too elaborated in the text and, moreover, it's claimed anyway that PT is not of much relevance for the experiments which are typically in the nonperturbative regime.
What I miss is a more coherent picture unifying all essential ingredients, i.e., perturbation theory, NRG, and experimental findings.

I am not sure if and how this could be improved but it would definitely help the paper as well as the readers.

Here I list various questions, comments, and suggestions for authors' consideration:
1-\beta-index missing in Eqs. (8), (17), and (18). Matrix elements m_A/B,n\sigma in Eqs. (17) and (18) not defined nor mentioned anywhere else.

2-Horizontal axes in Figs. 2, 3, and 4 are not in dimensionless units (in other graphs they are).
Very misleading/confusing caption in Fig. 2 ("U relative to the gap \Delta AND (?!) the bandwidth D") .
Problems with caption of Fig. 3: (top) - wrong?, grey lines nearly invisible in my printout, missing = in the last line.
Problems with caption of Fig. 4: I don't see any RED dashed lines, just some kind of orange rather close to the orange full line, moreover NO dashed line for large U mentioned in the caption. Reference to a mysterious logarithmic behavior, see also below.
Misprint in "bandwidth" between Eqs. (25) and (26).

3-General reservations to handling and presenting perturbative results.
Perturbation theory, unlike NRG, provides (semi)analytical results and, thus, the depiction and presentation conventions are somewhat different from purely numerical, e.g., NRG results. It's a good practice to present results as functions of a minimal number of dimensionless parameters, i.e., the potentional dimesionful prefactor is extracted and only dimensionless integrals are analyzed. In Eq. (23) this would correspond to writing (dimensionless by its nature) kappa^(2) as Gamma/Delta x dimensionless integral depending on two dimensionless variables U/Delta and D/Delta.
Furthermore, very much unlike NRG, the bandwidth is not the most important and thus the reference parameter. It's far more natural to choose as the energy unit the BCS gap Delta. Then the bandwidth can be easily sent to infinity which is a generic analytical approach also of broad experimental relevance. In fact, I have never seen D being fitted from the experiment (I don't say it's impossible, just not very common I guess), while Gamma, U, and Delta are routinely (attempted to be) extracted. You write yourselves that the narrow-band limit D<<U, where D is important, "is mostly of academic interest" [just above IV.C]. The integral I^(2) in Eq. (23) can be straightforwardly analytically calculated even for finite D and in the limit of D->infty one just gets a single dimensionless function of the dimensionless parameter U/Delta with a clear asymptotic behavior. For finite D, there is the D-dependent truncation for large values of the parameter. It can be intuitively wrapped into one single plot (a hopefully self-explanatory skeleton example attached); making it into a three panel figure (moreover for a fixed D) is in my view just a confusing overkill.
Similarly, just worse, for the fourth order. Here, you don't even present the formulas defining the integrals in Eq. (33). It's then quite hard (to try) to reproduce your results, which directly contradict the SciPost requirement "Provide sufficient details (inside the bulk sections or in appendices) so that arguments and derivations can be reproduced by qualified experts;" of the general acceptance criteria. Again, it would be nice to make the integrals dimensionless by pulling out 1/Delta^2 factor. That would decrease your (ideally small) pertubative correction by eight orders of magnitude from 10^11 to about 1000. Yet, it's still a big number and one might question what the status of the perturbative expansion actually is when the fourth order gives a factor about 1000 times bigger than the second one. Does this order of PT converge at all for infinite D? I can't answer those questions since you don't give any formulas but I would love to get the answers. You also mention some logarithmic correction (whose origin is completely unclear) which miraculously adds up with a 1/U^2 asymptotics of the 4b contribution to a constant plateau [just above IV.C.3]. Furthermore, you study asymptotics of large U>>D (i.e., in the narrow-band limit) which you claimed to be "mostly of academic interest" in just previous subsection. However, I can't see anywhere the far more interesting, generic, and thus relevant D->infty limit.
I believe that these issues should be addressed far more thoroughly to bring solid understanding of the perturbative behavior of the model.
Two more particular questions to this part:
i. Why the curve in Fig. 3f grows for small U while it plunges in Fig. 3e? I roughly understood that compared to the above panels c) and d) you basically multiply the curves by U^2 or U^4, respectively which should make them both fall down for small U, the one in f) even faster. Where am I wrong?
ii. In Fig. 5b you plot the NORMALIZED difference of kappa^(4). Why normalized, why this way? Would this quantity be of any experimental relevance (my answer is "absolutely not"). Experimentally meaningful quantity is just the unnormalized difference which you in fact consider in Fig. 15. Presentation of the normalized difference is in my view redundant if not misleading...

4-In V.A I don't understand the term "saturation", which is in the text related to the small-Gamma asymptotics while in the title it seems to be its opposite. I can't observe "a near perfect overlap of the curves", sorry, I don't understand what's meant by that.
Why the legend in Fig. 7(left) changed on the fly from U/Delta to U/D?
I kind of grasp the explanation of the saturation in Sec. V.B but why the curves for very small Gamma/U don't also saturate for large U?
Missing = signs in caption of Fig. 9.
Don't you mean Gamma cos(phi/2) in V.D.1? Otherwise, I completely miss the justification...

5-Sec. VI:
I suggest to cite PRL 108, 227001 (2012) by Luitz et al. in the context of relevance of SC-SIAM for modelling experiments since this was one of the main conclusions of that paper which precedes all the currently mentioned references.
I don't understand Fig. 13 and the explanations just below it (first two paragraphs). It's perhaps obvious to the NRG people but certainly not to a nonspecialized reader. Was the impurity density matrix in the figure calculated to for a finite applied magnetic field (since the occupations of spin up and down are dramatically different)? What's the value of the field? What happened to the red curve in the right panel? Could you describe what's seen in the figure before you start writing about things which are not there? I like the idea of the connection between the Kondo and SC-Anderson model and the different asymptotics of kappa in the two models. However, I would need a last sentence (here missing) to understand the implications of Eq. (49) for the J^2 behavior. Could you be more eloquent?

6-Finally, in the first line on p. 19 you write that "For devices with higher values of U/Delta...even larger shifts can be achieved...". Doesn't Fig. 15 show the opposite?

1-timely manuscript which makes contact to a very recent paper on the same question for the Kondo model (here the single-impurity Anderson model) as well as to experimental data
2-comprehensive presentation including perturbation theory (in the impurity-reservoir hybridization) and (accurate) numerical for results beyond the perturbative regime (which includes the parameter regime relevant to most experiments)
3-overall well written (for suggestions; see below)

The only weaknesses I found are of detailed nature. I will list them below.

In the manuscript the change of the impurity g-factor due to the coupling of the impurity to two (BCS) superconductors is analyzed theoretically in due detail. This is a question of current theoretical and experimental interest. The manuscript is directly related to recent theoretical and experimental publications. It clarifies the role of charge fluctuations (Kondo model versus single-impurity Anderson model) and discusses the dependence of the g-factor change (Knight shift) on the phase difference between the two supercondutors. I am confident that the present work is of current interest, clarifies a pressing reserach question, and will lead to follow-up publications. The presentation is overall very clear. The results are discussed in a transparent form. I can thus recommend publication. However, I stumbled accross a few detailed issues the authors might want to revise to even enhance the clarity of the presentation. I'll gives these below.

This is a list of suggestions. It should be obvious which ones are minor (e.g. typos or so) and which ones are more important.
1-page 4, first paragraph of Sect. IV: "...using the projector
operator approach AND a symbolic algebra system..."
2-Notation Eqs. (15), (17), (18): I suggest to use the bracket-notation also for the states appearing on the left hand sides of these equations. I was inintially confused by this switch of notation. This appears to be unnecessary.
3-page 6, right before Eq. (25): Speaking here and following Eq. (30) of U<<Delta is confusing as what is given is in fact the U=0 result. In particular, no corrections in oders of U/Delta are given.
4-Fig. 3: It woulld look nicer if there would be a small space between the "I" and the upper index "(4a/b)".
5-page 7, right below Fig. 3: What is meant by "...the same rule as in the previous section..."? The argument employing the intermediate states?
6-Fig. 4: I suspect that the gray lines again indicate U=Delta and U=D? Please clarify. I am confused by the caption. I can only see red dasehd lines showing the asymptotics at small U!?
7-page 9, last line: The sentence "The leading Γ^2 correction better captures the deviation from linearity..." is confusing. The Gamma^2 term IS the leading correction to linearity and thus must be included to capture any deviation from linearity. Furthermore, "...underestimate the renormalization in the intermediate Γ regime." What exactly is the "intermediate Gamma regime"?
8-page 10, second paragraph: I am puzlled by the discussion of "...clearly visible saturation effects...". I do not recognize any saturation in the two plots of Fig. 10 while the linearity in the corresponding variables as described in the text is obviuos.
9-Fig. 8 and last sentence of Sect. V.B: Is there a typo in the x-axis label? As I understood it D is always set to 1. I suspect that the x-axis label is U/Delta? In that case the last sentence of Sect. V.B appears to be meaningful as well (otherwise not).
10-Sect. V.D, bottom of page 11.: As the doublet-singlet quantum phase transition is not discussed in the present manuscript as reference might be useful.
11-Sect. V.D: Inclusing a single subsubsection "1...." is confusing. Just drop the "1....".
12-page 13: How do you know that the cos(2 phi) term is fourth order in Gamma? Please elaborate.
13 - Sect. V.F: I do not understand why the attempt to capture the Theta dependence of kappa in a phenomenological way (Eqs. (439-(46)) is meaningful or useful. Please elaborate.