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Exploring the vortex phase diagram of Bogoliubov-de Gennes disordered superconductors

by Bo Fan and Antonio Miguel Garcia Garcia

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Submission summary

Authors (as registered SciPost users): Antonio Miguel Garcia Garcia
Submission information
Preprint Link: scipost_202305_00036v2  (pdf)
Date submitted: 2023-09-10 18:02
Submitted by: Garcia Garcia, Antonio Miguel
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approach: Theoretical


We study the interplay of vortices and disorder in a two-dimensional disordered superconductor at zero temperature described by the Bogoliubov-de Gennes (BdG) self-consistent formalism for lattices of sizes up to 100X100 where the magnetic flux is introduced by the Peierls substitution. We model substantially larger lattice size than in previous approaches (36X36) which has allowed us to identify a rich phase diagram as a function of the magnetic flux and the disorder strength. For sufficiently weak disorder, and not too strong magnetic flux, we observe a slightly distorted Abrikosov triangular vortex lattice. An increase in the magnetic flux leads to an unexpected rectangular vortex lattice. A further increase in disorder, or flux, gradually destroys the lattice symmetry though strong vortex repulsion persists. An even stronger disorder leads to deformed single vortices with an inhomogeneous core. As the number of vortices increases, vortex overlap becomes more frequent. Finally, we show that global phase coherence is a feature of all these phases and that disorder enhances substantially the critical magnetic flux with respect to the clean limit with a maximum on the metallic side of the insulating transition.

Author comments upon resubmission

Dear Editor,
Thanks for forwarding the three referee reports. A detailed response has been submitted separately. We think we have addressed all referee comments and suggestions satisfactorily. Indeed, the three reports were constructive and have helped make the paper better.
We are enclosing an updated manuscript.
We hope that the paper is now suitable for SciPost.
Your Sincerely,
The authors.

List of changes

Following the referees comments and suggestions, we have made the following changes in no particular order:
1. We have carried out a careful profreading of the full manuscript.
2. The plots providence direct evidence of the existence of triangular vortex lattice in the clean limit have been moved from the appendix to section III of the main text.
3. In section V, we switched to the Ginzburg-Landau theory prediction of the profile of the order parameter inside the vortex to compare with the numerical result from the solutiosn of the BdG equations.
4. In section II, after the introduction of the model, we have clarified why the range of parameters we are interested in are quite different to those employed to describe the physics of Hofstedter superconductors.
5. We have added structure factor plots (Fig. 2, 3, 5, 7) in order to provide sharop evidence of the vortex latice structure. Details of the calculation are in appendix D.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2023-9-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202305_00036v2, delivered 2023-09-26, doi: 10.21468/SciPost.Report.7849


The authors have addressed of all my main comments. Concerning point 5, to clarify, what I meant is the correlation between the spatial distribution of the SC gap Δ and the disorder potential, for a fixed realization of the disorder. One expects that, as the authors claim, that the vortex cores are pinned to areas with strongest disorder. Figure R4 seems to suggest that this is indeed the case, at least for stronger disorder, but it is not clear from data presented in the manuscript. A plot of V(r) alongside the plots of Δ(r) already in the text would address my question. This seems especially important given the discussion by the authors of the need for a self-consistent BdG calculation, since they state that the disorder in Δ(r) does not follow the same distribution as the disorder in V(r).

Additionally, I would suggest another proof-reading of the text as many grammatical errors remain. For example, on page 15: "where it is observed a clear deformation" should instead read "where a clear deformation is observed." Another common mistake is the use of "the vortices position," which should either read "the vortex position" or "the position of vortices." Once these are fixed, I believe the manuscript will be acceptable for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
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Author:  Antonio Miguel Garcia Garcia  on 2023-09-27  [id 4018]

(in reply to Report 3 on 2023-09-26)
answer to question

See attached file with a detailed response to the referee comments and questions.



Anonymous Report 2 on 2023-9-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202305_00036v2, delivered 2023-09-12, doi: 10.21468/SciPost.Report.7814


See previous report


due to revisions previous weaknesses have been mostly removed but one, see report


Authors have convincingly replied to the issues I have raised in my previous report and have revised the manuscript accordingly. There is, however, still one minor point related to the correlation function in Fig. 13. While authors have provided a better definition of the correlation function they should specify the system size in the caption to Fig. 13. For a NxN lattice the maximum distance is N/sqrt(2) along the diagonal. For a 60x60 lattice this would correspond to a maximum distance of ~42 and the correlation function for larger distances, i.e. 42+d, should be the same than for 42-d. This is what I meant with 'periodicity' in my previous report. So I guess that authors have used probably 100x100 for Fig. 13 but this should be specified.

Requested changes

* Provide the system size in the caption to Fig. 13

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: good

Author:  Antonio Miguel Garcia Garcia  on 2023-09-27  [id 4017]

(in reply to Report 2 on 2023-09-12)

See attached file with a detailed response to the referee comments and questions.



Anonymous Report 1 on 2023-9-11 (Invited Report)


In my view the authors have answered the comments and suggestions and the new version is ready for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Antonio Miguel Garcia Garcia  on 2023-09-27  [id 4016]

(in reply to Report 1 on 2023-09-11)

We thank the referee for accepting the paper for publication and the previous comments and suggestions that helped make the paper better.

Login to report


Anonymous on 2023-09-10  [id 3972]


  • Some references are still lacking a DOI (e.g., [2,9,11,16,20,22,26,43,45,66,67,71-78]).
  • Why not use the SciPost style file?

Anonymous on 2023-09-27  [id 4015]

(in reply to Anonymous Comment on 2023-09-10 [id 3972])
answer to question

Thanks for pointing out the missing DOI's. In the updated version of the manuscript, we have added all the DOI's that we could find. We have also switched to SciPost style.