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Impurity Knight shift in quantum dot Josephson junctions

by Luka Pavešić, Marta Pita-Vidal, Arno Bargerbos, Rok Žitko

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Submission summary

Authors (as registered SciPost users): Luka Pavesic · Rok Žitko
Submission information
Preprint Link:  (pdf)
Data repository:
Date submitted: 2023-03-02 17:13
Submitted by: Pavesic, Luka
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Experiment
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Experimental, Computational


Spectroscopy of a Josephson junction device with an embedded quantum dot reveals the presence of a contribution to level splitting in external magnetic field that is proportional to $\cos \phi$, where $\phi$ is the gauge-invariant phase difference across the junction. To elucidate the origin of this unanticipated effect, we systematically study the Zeeman splitting of spinful subgap states in the superconducting Anderson impurity model. The magnitude of the splitting is renormalized by the exchange interaction between the local moment and the continuum of Bogoliubov quasiparticles in a variant of the Knight shift phenomenon. The leading term in the shift is linear in the hybridisation strength $\Gamma$ (quadratic in electron hopping), while the subleading term is quadratic in $\Gamma$ (quartic in electron hopping) and depends on $\phi$ due to spin-polarization-dependent corrections to the Josephson energy of the device. The amplitude of the $\phi$-dependent part is largest for experimentally relevant parameters beyond the perturbative regime where it is investigated using numerical renormalization group calculations. Such magnetic-field-tunable coupling between the quantum dot spin and the Josephson current could find wide use in superconducting spintronics.

Current status:
Has been resubmitted

Author comments upon resubmission

We would like to thank the editor and both referees for their very constructive feedback. We feel that it helped us greatly improve the manuscript.

Taking the suggestion from Referee #2, we changed the way the theoretical and experimental results are related. The measurements are now presented at the very beginning and serve as the motivation for the calculations that follow. Furthermore, we have significantly improved the presentation and the discussion of results in the perturbation-theory (PT) sections. In the section on fourth-order PT, the focus is now on the physical quantities as opposed to the properties of each contribution which by themselves do not carry much physical information. We only managed to solve the I^(4) integrals analytically in U=0 and U->\infty limits. To obtain the D>>U,Delta limit one would have to analytically solve the full integral and then calculate the limit of large D, which seems unfeasible. Nevertheless, it seems that U=0 results allow to estimate the renormalization factor kappa^4 for finite U << Delta, while the U=\infty result is a reasonable approximation for the Delta < U < D regime.

The perturbation theory approach is valuable as it gives physical intuition into the type of processes that produce the corrections. In our case this gives the important insight that the coherent fourth-order pair transfer is the culprit for phi-dependent kappa. This is thus an important part of the manuscript and needs to be included. On the other hand, the numerical calculations (using NRG) provide accurate numerical results with no parameter restrictions. Comparing the two (like in Fig. 7) quantifies the regime where understanding the problem using simpler low-order processes is valid and shows where higher-order processes become important.

We hope that we have improved the clarity of the manuscript and that it now fulfils all criteria for acceptance.

List of changes

- Changed the wording in the abstract and introduction to better reflect the fact that the experimental data are used as motivation for theoretical developments.
- Moved the Experimental evidence section to the front.
- Explicitly wrote out the matrix elements in and around Eq. (20), (21).
- Reduced Fig. 3 (previous Fig. 2) to a single panel with dashed lines for limits.
- Rewrote the discussion in Sec. V B. The integral is now expressed with dimensionless parameters (eq. (28)).
- Clarified the matrix elements in Eq. (39). The full expressions of I^(4a) and I^(4b) are very long and we feel writing them explicitly would not contribute to the manuscript. They are however available in a Mathematica Notebook.
- Rewrote the discussion and expressions in Sec. V C.
- Removed panels c-f of Fig. 4. Their only purpose was to confirm the agreement of the analytically obtained limits to the numerical integration.
- Fixed colors in Fig. 5.
- Fig. 6(b) now shows the absolute difference between \kappa at \phi=0 and \phi=\pi.
- Cleared up the confusion regarding the word "saturation" in Sec. VI A and regarding Fig. 8.
- Replaced the reference to the singlet subgap state in Sec. VI D. Now we refer to spin screening in the doublet ground state.
- Removed the subsubsection VI D 1.
- Fixed the cos(phi) -> cos(phi/2) typo.
- Sec. VI E; cleared up the reasoning for the order of higher harmonics.
- Removed the discussion of \kappa(\theta) in Sec. VI F.
- Added a reference to PRL 108, 227001 (2012) in Sec. VII A.
- Added the P_up line to Fig. 14(b).
- Improved the explanation of the physical relevance of different P in Sec. VII B.
- Added the expression for the correction factor necessary for the Kondo model, below eq. (51).
- Added a few words in Sec. VIII to clarify Fig. 15.

Submission & Refereeing History

Resubmission 2212.07185v3 on 19 May 2023

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Resubmission 2212.07185v2 on 2 March 2023

Reports on this Submission

Anonymous Report 1 on 2023-4-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2212.07185v2, delivered 2023-04-05, doi: 10.21468/SciPost.Report.7010


I will basically just comment on the changes made in the updated version.

At the first sight, it seemed that authors made the required changes and the updated manuscript is now fine. However, when plunging more into details especially in the analytical section V on the perturbative approach I have to my disappointment discovered that only cosmetic changes were performed and most of the issues remained. Some of my comments were just completely ignored and I still find the presented results very problematic for several reasons.

First, authors insist on not revealing the integrals for the fourth order contribution and refer to the Mathematica notebook instead. I don't consider Mathematica notebook to be an equivalent replacement of a formula, even if potentially long and complicated but OK. So I went to the repository and downloaded that half-a-gigabyte large zip archive. Unfortunately, my computer somehow hasn't managed to unpack the beast (tried twice) so that I have not been able to see the formulas anyhow yet. My computer is not particularly old or weak so I don't know what to take from that.

Second, the wide band limit is claimed in general inaccessible by the authors but I cannot agree with that. It's likely and would be very useful to show it explicitly that the sum of the two fourth-order contributions has a finite limit when D goes to infinity as it was clearly demonstrated in Eqs. (43) and (44) for the U=0 case. The integration can be performed numerically even for infinite integration limits as long as the integrand decays fast enough. That's exactly the reason why I would need to see the formulas... Instead, the "academic" U>>D limit results are presented and proclaimed in the answer to be "a reasonable approximation for the Delta<U<D regime", although this is not true even in the second order, where it gives qualitatively different results, compare Eq. (35) with (37) .

There are other questions concerning the validity and usefulness of the presented analytical approach and results. For example, particle-hole symmetric case is always considered in Sec. V. Does the limit U->infty then make any physical sense? I believe that U=infty is a useful approximation under suitable conditions, but then one typically considers finite value of epsilon (single-particle level energy). Sending both energies to infty in my view gives just another purely academic answer with no practical implications.

I am not sure, but isn't the phi-dependent part of kappa given by the magnetic field derivative of the critical current through the junction? If so, it would be useful to explicitly mention it. Then another question arises concerning results in Eq. (42). The critical current of a noninteracting resonant level (epsilon=U=0) diverges within the cotunneling approximation (easily seen - the critical current as a function of epsilon is given by a Lorentzian curve; at the resonance one gets 1/Gamma, i.e. the perturbation theory is singular at this point). This points at a possible issue with the order of various limits (U=0, epsilon=0, B=0) and should be handled with big care...

I don't understand the curves in Figs. 4 and 5 for small U. According to Eq. (43) U=0 limit gives for D=10^4 roughly -275 for the a-contribution and 355 for the b-contribution. Their sum (the physically relevant result) is about 80. I am not asking about the logarithm of a negative number in Fig. 4a (I guess the absolute value was taken), but I really don't understand how the summed up contribution (about 80) in Fig. 5a can be above the partial ingredients (around 300) [I am aware of the multiplication by x^2 and logarithmic axes but still, it should be below the two].

BTW, it would be good to explicitly refer to the corresponding analytical expressions for large and small U in the caption of Fig. 5, they are not immediately obvious. What is "r" in Eq. (45)?

I find a bit outrageous the uncorrected statement about the "logarithmic behavior, which is best seen on the log-linear scale in Fig. 5b" (top of p. 10). The authors refer to integrals which were sadly never revealed anywhere so that they are hard to appreciate. Moreover, one should see a straight line in the range (1,10^4) but there are not straight lines (blue and yellow).

I do agree with the authors that "the perturbation theory approach is valuable as it gives physical intuition into the type of processes that produce the corrections. In our case this gives the important insight that the coherent fourth-order pair transfer is the culprit for phi-dependent kappa".

Therefore I strongly suggest:
1. (Physics) Elucidate what is the relation of the phi-dep. kappa and the supercurrent.
2.(Math, Formal) Correct and enhance the analytical perturbative results. In particular, please provide in an accessible way the formulas behind the fourth order and perform their reliable analysis.

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