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Collective modes in superconductors including Coulomb repulsion
by Joshua Althüser, Götz S. Uhrig
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Joshua Althüser · Götz S. Uhrig |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2503.05494v1 (pdf) |
| Date submitted: | March 10, 2025, 11:09 a.m. |
| Submitted by: | Joshua Althüser |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We numerically study the collective excitations present in isotropic superconductors including a screened Coulomb interaction. By varying the screening strength, we analyze its impact on the system. We use a formulation of the effective phonon-mediated interaction between electrons that depends on the energy transfer between particles, rather than being a constant in a small energy shell around the Fermi edge. This justifies considering also rather large attractive interactions. We compute the system's Green's functions using the iterated equations of motion (iEoM) approach, which ultimately enables a quantitative analysis of collective excitations. For weak couplings, we identify the well-known amplitude (Higgs) mode at the two-particle continuum's lower edge and the phase (Anderson-Bogoliubov) mode at $\omega = 0$ for a neutral system, which shifts to higher energies as the Coulomb interaction is switched on. As the phononic coupling is increased, the Higgs mode separates from the continuum, and additional phase and amplitude modes appear, persisting even in the presence Coulomb interactions.
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
Report #2 by Anonymous (Referee 2) on 2025-6-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.05494v1, delivered 2025-06-16, doi: 10.21468/SciPost.Report.11408
Strengths
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The paper considers an interesting and important subject: collective modes of strongly-coupled superconductors with realistic momentum-dependent interactions.
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Results are sound and are qualitatively reasonable, to the best of my judgement.
Weaknesses
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There is very little on the way of interpretation of the obtained results. For example, are the extra Higgs modes discovered by the authors correspond to radial (zero angular momentum?) bound-states of the two-particle problem with the same momentum-dependent pairing potential? Such two-particle problem can be solved analytically or numerically and results compared.
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The very similar situation was discovered 60+ years ago by Bardasis and Schrieffer. The authors mentioned them in one sentence. This is certainly not enough. They should articulate similarities and differences between their findings and those of BS much more explicitly.
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I find the use of static screening interactions (3) somewhat puzzling. The conventional treatment, known as "Tolmachev logarithm", calls for renormalizing the Coulomb line with the Cooper ladder, leading to: V_C --> V_C/(1+V_C\rho\log(E_F/\omega_D)) \approx 1/\rho*\log(...)
The authors should explain what makes them deviate from this paradigm (usually thought to explain why phonon effect can ever overcome the Coulomb one).
Report
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #1 by Anonymous (Referee 1) on 2025-5-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2503.05494v1, delivered 2025-05-28, doi: 10.21468/SciPost.Report.11291
Strengths
- It is a creative paper in its physically justified choice of the interaction with quite nontrivial result
- The results are correct - I was able to verify the shift of the Higgs mode myself using somewhat different method.
- The authors provide an important insight for the origin of the effect which is nicely illustrated in Fig. 5
Weaknesses
There is only one weakness here:
- The manuscript does not offer anything substantially new conceptually. In fact, the work uses essentially the same idea - momentum dependent interaction - as the one that has been put forward in a paper by Barankov and & Levitov [arXiv: 0704.1292] and thoroughtly forgotten since then. Specifically Barankov and Levitov have also found that momentum dependent interaction lead to the shift of the Higgs mode below the single-particle threshold. I believe the authors should cite and discuss the results of that work since, as I mentioned above, conceptually it has quite a bit of overlap with the present study although the methodology is a bit different.
Report
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
We thank the Referee for their time and kind and positive report and for making us aware of the work by Barankov and Levitov.
We agree that the detachment of the Higgs mode from the quasiparticle continuum has been observed by these colleagues previously in their preprint. Consequently, we added a brief discussion of their results as well.
We would like to highlight, that we propose additional modes beyond the finding of Barankov and Levitov, namely the existence of Higgs (and phase) modes beyond the primary ones for stronger interactions. We think that this represents an important novel finding.
We would like thank the Referee again for the interesting reference, thereby improving our manuscript by connecting it to previous research.
Author: Joshua Althüser on 2025-06-23 [id 5589]
(in reply to Report 2 on 2025-06-16)We thank the Referee for their time and kind and positive report as well as for the constructive suggestions. Let us address the points individually:
Our calculations are done at zero angular momentum and at zero center-of-mass momentum. We have added a few explanatory sentences to make this clearer. We do not believe that there is an analytical solution to this problem because the effective interaction is not a density-density interaction in real space. This assessment is supported by the result from the reference (Barankov and Levitov [arXiv: 0704.1292]) provided by the other Referee, in which also a detachment of the Higgs mode from the lower edge of the two-particle continuum has been observed. This finding was obtained by numerical means as well.
We agree that the results of Bardasis and Schrieffer deserve more discussion and have added a paragraph accordingly. The important distinction between the results by Bardasis and Schrieffer and ours in the manuscript consists in the angular momentum: We performed calculations at zero angular momentum L=0, while the Bardasis-Schrieffer modes hinge on L \neq 0.
The usage of the screening (3) is a standard procedure known as the Thomas-Fermi screening. This type of expression arises from chains of bubble diagrams (cf. Fig. 1). In fact, this approach has been used before in the context of superconductors (cf. Simonato, Katsnelson, and Rösner in https://doi.org/10.1103/PhysRevB.108.064513). We further note that for a generic screening strength, we reproduce the results by Tolmachev, and Morel and Anderson. Hence, the renormalization of the interaction by the Cooper ladder. We added a paragraph below Eq. (3) discussing these connections.
We would like thank the Referee again for their aid in improving our manuscript by encouraging us to provide clearer explanations of our assumptions and calculations.