SciPost Submission Page
A Large-$N$ Approach to Magnetic Impurities in Superconductors
by Chen-How Huang, Alejandro M. Lobos, Miguel A. Cazalilla
Submission summary
| Authors (as registered SciPost users): | Chen-How Huang |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202409_00023v1 (pdf) |
| Date submitted: | 2024-09-19 22:43 |
| Submitted by: | Huang, Chen-How |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Quantum spin impurities coupled to superconductors are under intense investigation for their relevance to fundamental research as well as the prospects to engineer novel quantum phases of matter. Here we develop a large-$N$ mean-field theory of a strongly coupled spin-$\tfrac{1}{2}$ quantum impurity in a conventional $s$-wave superconductor. The approach is benchmarked against Wilson's numerical renormalization group (NRG). While the large-$N$ method is not applicable in the weak-coupling regime where the Kondo temperature $T_K$ is smaller than the superconducting gap $\Delta$, it performs very well in the strong coupling regime where $T_K \gtrsim \Delta$, thus allowing us to obtain a reasonably accurate description of experimentally relevant quantities. The latter includes the energy of the Yu-Shiba-Rusinov subgap states, their spectral weight, as well as the local density of continuum states. The method provides a reliable analytical tool that complements other perturbative and non-perturbative methods, and can be extended to more complex impurity models for which NRG may not be easily applicable.
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 multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1-High level of detail, all steps of the derivation are very clear.
2-Method seems to work well in the singlet regime.
Weaknesses
1-Limitation to particle-hole symmetric cases.
2-Limitation to strong-coupling (singlet) regime.
3-Unclear whether the method can indeed be applied to more complex situations.
Report
This work shows that a large-N approach for solving the single-impurity Anderson model with a superconducting bath gives reasonably accurate results (as benchmarked against the NRG solution). Since this problem can be approximately solved with a number of simple techniques, e.g. exact diagonalisation in the "zero-bandwidth" limit on a two-site cluster and the refinements of this approach (such as the "surrogate model solver"), atomic limit solution and its various generalizations, or standard perturbation theory (at least for the singlet state), which all provide reasonable or even very good approximations, the addition of large-N solver to this list does not represent by itself a significant advance, especially because the approach is limited to the particle-hole symmetric and strong-coupling situation. The method is potentially applicable to more complex models, and if this were demonstrated by the authors, the significance of this work would be greatly improved. Therefore I would strongly suggest that the authors consider extending this manuscript with some generalizations of the model (e.g. to the two-lead situation with phase-bias, as in the Josephson-Anderson model, or the inclusion of the Zeeman term). If the method provides accurate description also in those situations, the potential impact of the work would be much bigger. Another possible improvement would be inclusion of fluctuations to go beyond the saddle-point approximation and perhaps capture the properties of the doublet state.
This being said, I actually like the manuscript, it is written very clearly and the presentation is very good, I also like the great level of detail. To the best of my ability to judge, it is technically correct. Therefore I would not object to this work being published even in its present form.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
The authors present a large-N mean field approach to the Kondo model for a magnetic impurity in a superconductor. They show results for the subgap states position and the local density of states which compare reasonably well with NRG calculations in the limit of large Kondo temperature.
In my opinion the approach and the results presented in this manuscript are sound and deserve to be published in Scipost. The manuscript is written in a rather pedagogical way, which I find very valuable.
The only point which I find a bit obscure is the choice of the band-width for the comparison of large-N and NRG calculations. As indicated in Eq. (25), the authors require that the density of states at the Fermi level, \rho_0, should be the same in
both calculations, i.e. \rho_0 = 1/(\pi t) = 1/2D, where t is the hopping element in the effective TB model and D is the NRG band-width. Thus, I would have expected that for the comparison they choose a ratio D/t such that \rho_0\Delta is the same in both calculations. However, taking the values indicated in Fig. 4(a) I find \rho_0\Delta= 0.0005 in the NRG calculation, and \rho_0\Delta = 0.005/\pi for the large-N one. There might be some misunderstanding from my side but in any case it would be worth that the authors could clarify this issue in their manuscript.
The authors might also consider giving a reference to some recent STM experimental works where YSR were detected and correlated with the Kondo effect in the normal state (see Communications Physics 6, 214 (2023) and references there in).
Notice also a typo in the sentence "We speculate that this is feature...".
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)