Impurity Knight shift in quantum dot Josephson junctions

We would like to again thank the referee for constructive feedback, which has helped us to remove the remaining issues with the manuscript.

Regarding the Zenodo repository:
We have not found anything wrong with the archive, the compatibility issue might indeed be related to the file size or with the use of tgz format, which might be problematic on some computer platforms. For this new version, we provide a new set of files in a more commonly used zip format, one per figure, thereby also fixing the problem of large files, since only some datasets are large. The Mathematica notebooks are, in fact, small files.

Regarding the manuscript:
We thoroughly reworked Section V C on the fourth-order perturbation calculations. The integrals are now explicitly given in terms of quasiparticle energies in a somewhat compact form; we apologise for not providing those in the body of the manuscript in the previous versions. Also we would like to thank the reviewer’s insights which has helped us to correct a sign issue in the definition of the fourth-order correction (relative sign of a vs. b terms). The main results are not affected by this sign change in a significant way since the cos(phi) term had the correct sign. Nevertheless, fixing the sign issue is important because the phi-independent fourth-order correction actually changes sign at U somewhat above 2Delta. In fact, this sign change can be observed in the change of curvature in the NRG results shown in Fig. 6a (new numbering in revised version): the curves for small U curve downwards (when kappa^4 is negative), while they curve upwards at large U (kappa^4 > 0). The fourth-order results we report are now correct and the perturbation matches the low-Gamma numerics slightly better (Fig. 5, new numbering).

We removed the expressions for the U->infty limit at finite bandwidth, as we agree that they perhaps indeed do not contribute all that much to the physical understanding. In their place, there is now a discussion of the integrals in the wide-band limit. The two fourth-order contributions are no longer considered separately, as this way of splitting the processes comes from the mathematical formulation of the perturbation theory and it is thus somewhat arbitrary. Instead, we chose to discuss the phi-dependent and the phi-independent processes separately. We were not able to find closed-form expressions for the integrals, but we present the numerical results in the form of a product of the leading U/Delta-dependence multiplied by a correction function of order 1 (for U/Delta in the usual range). These correction functions are shown in the new Fig. 4. We also remark that to obtain well convergent integrals for numerical evaluation, we symmetrized the integrand with respect to one of the integration variables to cancel out a problematic asymmetric slowly converging term.

Regarding the U->infty limit (such that epsilon/U remains constant): While academic, it is commonly used in the context of mapping between the SIAM and Kondo models (Schrieffer-Wolff transformation) and thus very important in the theory of quantum impurity models. Here epsilon/U is fixed so that the quasiparticle scattering phase shift remains constant when taking the limit. But we agree that this limit is less relevant as regards the analysis of experimental results.

We added a brief discussion (Sec. VII A) on the relation between kappa and Josephson energy, which is itself proportional to the supercurrent.
We agree with the referee that one could obtain kappa by taking the field derivative of the critical supercurrent. However, in our formulation kappa is directly calculated and therefore we avoid any problems arising from taking derivatives of potentially diverging quantities. In particular, kappa does not appear to be singular for epsilon=U=B=0.

We hope that the improvements to the discussion of fourth-order perturbation theory are satisfactory.