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Variational solution of the superconducting Anderson impurity model and the band-edge singularity phenomena
by Teodor Iličin, Rok Žitko
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Rok Žitko |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2503.18902v1 (pdf) |
| Data repository: | https://zenodo.org/records/15089183 |
| Date submitted: | March 26, 2025, 12:33 p.m. |
| Submitted by: | Rok Žitko |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We propose a set of variational wavefunctions for the sub-gap spin-doublet and spin-singlet eigenstates of the particle-hole symmetric superconducting Anderson impurity model. The wavefunctions include up to two Bogoliubov quasiparticles in the continuum which is necessary to correctly capture the weak-coupling asymptotics in all parameter regimes. The eigenvalue problems reduce to solving transcendental equations. We investigate how the lowest singlet state evolves with increasing charge repulsion $U$, transitioning from a proximitized state (a superposition of empty and doubly occupied impurity orbitals, corresponding to an Andreev bound state) to a local moment that is Kondo screened by Bogoliubov quasiparticles (Yu-Shiba-Rusinov state). This change occurs for $U = 2\Delta$, where $\Delta$ is the BCS gap. At this point, the band-edge effects make the eigenenergy scale in a singular way as $\Gamma^{2/3}$, where $\Gamma$ is the hybridization strength. Away from this special point, regular $\Gamma$-linear behavior is recovered, but only for $\Gamma \lesssim (U/2-\Delta)^2/\Delta$. The singular behavior thus extends over a broad range of parameters, including those relevant for some quantum devices in current use. The singular state is an equal-superposition state with maximal fluctuations between the local impurity charge configurations. Accurately capturing the band-edge singularity requires a continuum model, and it cannot be correctly described by discrete (truncated) models such as the zero-bandwidth approximation or the superconducting atomic limit. We determine the region of parameter space where the second spin-singlet state exists: in addition to the whole $U<2\Delta$ ABS region, it also includes a small part of the $U>2\Delta$ YSR region for finite values of $\Gamma$, as long as some ABS component is admixed.
Author indications on fulfilling journal expectations
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- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
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- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-6-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.18902v1, delivered 2025-06-06, doi: 10.21468/SciPost.Report.11352
Report
I think that this paper is well motivated and well placed in the context of previous variational approaches, as is detailed convincingly in the introduction and also in Sec. VIII. The ansatz is made plausible, and the variational optimization and the results derived are described very precisely. Results from the analytical and numerical evaluation are favorably compared with NRG data. The discussion of the band-edge singularity, i.e., the NRG result, Fig. 8, and the analytical result Eq. 42 is particularly interesting.
Concluding, this is a strong paper that should be published.
There are some optional and minor points the authors might take into account:
One could try to improve the very technical presentation, in particular in Sec. III B, by shifting some details to the appendix.
The dark green color in Fig. 6a is hard to distinguish from black.
In the sentence above Eq. 33, one may add "asymptotically, for Gamma/Delta -> 0, ..." for clarity.
It should be explicitly indicated (in the text and in the figure captions), which method is used (variational wave function or NRG) to get the results shown in Figs. 7 and 8, and how the heat map is constructed in practice.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 1) on 2025-6-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.18902v1, delivered 2025-06-06, doi: 10.21468/SciPost.Report.11349
Strengths
Weaknesses
Report
quasiparticles appearing in the correlated (Anderson-type) quantum impurity
coupled to the conventional (BCS-type) superconductor. They propose variational
wave-function approach to describe both, the Andreev or Yu-Shiba-Rusinov bound
states, which is valid for the arbitrary set of model parameters: Coulomb
repulsion (U), pairing gap ($\Delta$), hybridization strength ($\Gamma$). Specifically,
they focus on the half-filled impurity in the regime, U $\sim 2 \Delta$, where
changeover of the bound states occurs.
Details concerning construction of the doublet and singlet configurations are
presented in Sec. II - technicalities are convincingly discussed and numerical
results satisfactorily agree with the unbiased NRG calculations (Sec. IV).
Next, the authors analyze properties of their variational wave-functions,
inspecting matrix elements of the low-lying eigenstates (Sec. V). In particular,
they emphasize the maximal fluctuations between different charge-number states
in the case U = 2 $\Delta$. Finally the spatial profiles are investigated through
the Fourier-transformed momentum/energy resolved wavefunctions. Numerical
results (Fig. 13) indicate that upon increasing the hybridization strength
(Gamma) the quasiparticles tend to be more and more localized. In Sec. VIII
the authors confront their procedure with other methods, Refs. [26, 6, 37]
and clarify their limitations and/or complementary character.
In my opinion, the method proposed by T. Ilicin and R. Zitko could be of
potential use for studying various superconducting nanohybrid structures.
The manuscript is clearly written, analytical expressions seem to be
correct, validity of the computational data is carefully checked by
comparing them to NRG and other available methods. I thus recommend
the paper for publication in its present form.
Requested changes
None
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)