On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

  • Ilias Chenn
  • Panayotis Smyrnelis
  • Israel Michael Sigal
Article
  • 31 Downloads

Abstract

We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per fundamental cell. We also revisit the existence proof for the Abrikosov vortex lattices, streamlining some arguments and providing some essential details missing in earlier proofs for a single magnetic flux quantum per a fundamental cell.

Keywords

Magnetic vortices Superconductivity Ginzburg-Landau equations Abrikosov vortex lattices Bifurcations 

Mathematics Subject Classification (2010)

35Q56 

Notes

Acknowledgments

It is a pleasure to thank Max Lein for useful discussions and the anonymous referees for reading carefully the manuscript and many useful remarks and suggestions. The first author would like to thank Dmitri Chouchkov for useful discussions. The first and third authors’ research is supported in part by NSERC Grant No. NA7901. During the work on the paper, they enjoyed the support of the NCCR SwissMAP. The first author was also supported by the NSERC CGS program. The second author (P. S.) was partially supported by Fondo Basal CMM-Chile and Fondecyt postdoctoral grant 3160055.

References

  1. 1.
    Abrikosov, A.A.: On the magnetic properties of superconductors of the second group. J. Explt. Theoret. Phys. 32, 1147–1182 (1957)Google Scholar
  2. 2.
    Aftalion, A., Serfaty, S.: Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c2. Selecta Math. (N.S.) 13, 183–202 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Almog, Y.: On the bifurcation and stability of periodic solutions of the Ginzburg-Landau equations in the plane. SIAM J. Appl. Math. 61, 149–171 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Almog, Y.: Abrikosov lattices in finite domains. Commun. Math. Phys. 262, 677–702 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barany, E., Golubitsky, M., Turski, J.: Bifurcations with local gauge symmetries in the Ginzburg-Landau equations. Phys. D 56, 36–56 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chapman, S.J.: Nucleation of superconductivity in decreasing fields. European J. Appl. Math. 5, 449–468 (1994)MathSciNetMATHGoogle Scholar
  7. 7.
    Chapman, S.J., Howison, S.D., Ockedon, J.R.: Macroscopic models of superconductivity. SIAM Rev. 34, 529–560 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chouchkov, D., Ercolani, N.M., Rayan, S., Sigal, I.M.: Ginzburg-Landau equations on Riemann surfaces of higher genus. arXiv:1704.03422 (2017)
  9. 9.
    Du, Q., Gunzburger, M.D., Peterson, J.S.: Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34, 54–81 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dubrovin, D.A., Fomenko, A.T., Novikov, S.P.: Modern geometry – methods and applications. Part I. The geometry of sufraes, transformation groups, and fields. 2nd Edition. Springer-Verlag, Berlin (1984)MATHGoogle Scholar
  11. 11.
    Dutour, M.: Phase diagram for Abrikosov lattice. J. Math. Phys. 42, 4915–4926 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dutour, M.: Bifurcation vers l état dAbrikosov et diagramme des phases. Thesis Orsay . arXiv:math-ph/9912011
  13. 13.
    Eilenberger, G., Zu, A.: Theorie der periodischen Lösungen der GL-Gleichungen für Supraleiter 2. Z. Physik 180, 32–42 (1964)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fournais, S., Helffer, B.: Spectral methods in surface superconductivity. progress in nonlinear differential equations and their applications, Vol 77. Birkhäuser, Boston (2010)Google Scholar
  15. 15.
    Gustafson, S.J., Sigal, I.M.: Mathematical concepts of quantum mechanics. Springer, Berlin (2006)MATHGoogle Scholar
  16. 16.
    Gustafson, S.J., Sigal, I.M., Tzaneteas, T.: Statics and dynamics of magnetic vortices and of Nielsen-Olesen (Nambu) strings. J. Math. Phys. 51, 015217 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jaffe, A., Taubes, C.: Vortices and monopoles: structure of static gauge theories. Progress in Physics 2. Birkhäuser, Boston (1980)MATHGoogle Scholar
  18. 18.
    Kleiner, W.H., Roth, L.M., Autler, S.H.: Bulk solution of Ginzburg-Landau equations for type II superconductors: upper critical field region. Phys. Rev. 133, A1226—A1227 (1964)ADSCrossRefMATHGoogle Scholar
  19. 19.
    Lasher, G.: Series solution of the Ginzburg-Landau equations for the Abrikosov mixed state. Phys. Rev. 140, A523—A528 (1965)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Odeh, F.: Existence and bifurcation theorems for the Ginzburg-Landau equations. J. Math. Phys. 8, 2351–2356 (1967)ADSCrossRefGoogle Scholar
  21. 21.
    Ovchinnikov, Y.N.: Structure of the supercponducting state near the critical fiel H c2 for values of the Ginzburg-Landau parameter κ close to unity. JETP 85(4), 818–823 (1997)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Rubinstein, J.: Six Lectures on Superconductivity. Boundaries, interfaces, and transitions (Banff, AB, 1995), 163–184, CRM Proc. Lecture Notes, 13, Amer. Math. Soc., Providence, RI (1998)Google Scholar
  23. 23.
    Sandier, E., Serfaty, S.: Vortices in the magnetic ginzburg-landau model. Progress in nonlinear differential equations and their applications, vol. 70. Birkhäuser, Boston (2007)MATHGoogle Scholar
  24. 24.
    Sigal, I.M.: Magnetic Vortices, Abrikosov Lattices and Automorphic Functions, in Mathematical and Computational Modelling (With Applications in Natural and Social Sciences, Engineering, and the Arts). Wiley, New York (2014)Google Scholar
  25. 25.
    Takáč, P.: Bifurcations and vortex formation in the Ginzburg-Landau equations. Z. Angew. Math. Mech. 81, 523–539 (2001)MathSciNetMATHGoogle Scholar
  26. 26.
    Tzaneteas, T., Sigal, I.M.: Abrikosov lattice solutions of the Ginzburg-Landau equations. Contem. Math. 535, 195–213 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Tzaneteas, T., Sigal, I.M.: On Abrikosov lattice solutions of the Ginzburg-Landau equations. Math. Model. Nat. Phenom. 8(5), 190–205 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Ilias Chenn
    • 1
  • Panayotis Smyrnelis
    • 2
  • Israel Michael Sigal
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Centro de Modelamiento Matemático (UMI 2807 CNRS)Universidad de ChileSantiagoChile

Personalised recommendations