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Communications in Mathematical Physics

, Volume 337, Issue 1, pp 191–224 | Cite as

Lack of Diamagnetism and the Little–Parks Effect

  • Søren Fournais
  • Mikael Persson Sundqvist
Article

Abstract

When a superconducting sample is submitted to a sufficiently strong external magnetic field, the superconductivity of the material is lost. In this paper we prove that this effect does not, in general, take place at a unique value of the external magnetic field strength. Indeed, for a sample in the shape of a narrow annulus the set of magnetic field strengths for which the sample is superconducting is not an interval. This is a rigorous justification of the Little–Parks effect. We also show that the same oscillation effect can happen for disc-shaped samples if the external magnetic field is non-uniform. In this case the oscillations can even occur repeatedly along arbitrarily large values of the Ginzburg–Landau parameter κ. The analysis is based on an understanding of the underlying spectral theory for a magnetic Schrödinger operator. It is shown that the ground state energy of such an operator is not in general a monotone function of the intensity of the field, even in the limit of strong fields.

Keywords

Unit Disc Ground State Energy Trial State Landau Equation Lower Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Avron J., Herbst I., Simon B.: Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45(4), 847–883 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Berger J., Rubinstein J.: On the zero set of the wave function in superconductivity. Commun. Math. Phys. 202(3), 621–628 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berger M.S., Chen Y.Y.: Symmetric vortices for the Ginzburg–Landau equations of superconductivity and the nonlinear desingularization phenomenon. J. Funct. Anal. 82(2), 259–295 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Comte M., Mironescu P.: A bifurcation analysis for the Ginzburg–Landau equation. Arch. Ration. Mech. Anal. 144(4), 301–311 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Comte M., Sauvageot M.: On the Hessian of the energy form in the Ginzburg–Landau model of superconductivity. Rev. Math. Phys. 16(4), 421–450 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dombrowski N., Raymond N.: Semiclassical analysis with vanishing magnetic fields. J. Spectr. Theory 3(3), 423–464 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Erdős L.: Dia- and paramagnetism for nonhomogeneous magnetic fields. J. Math. Phys. 38(3), 1289–1317 (1997)CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Fournais S., Helffer B.: On the third critical field in Ginzburg–Landau theory. Commun. Math. Phys. 266(1), 153–196 (2006)CrossRefADSzbMATHMathSciNetGoogle Scholar
  9. 9.
    Fournais, S., Helffer, B.: Strong diamagnetism for general domains and application. Ann. Inst. Fourier (Grenoble) 57(7), 2389–2400 (2007). Festival Yves Colin de VerdièreGoogle Scholar
  10. 10.
    Fournais S., Helffer B.: On the Ginzburg–Landau critical field in three dimensions. Commun. Pure Appl. Math. 62(2), 215–241 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Fournais S., Helffer B.: Spectral Methods in Surface Superconductivity. Progress in Nonlinear Differential Equations and Their Applications, vol. 77. Birkhäuser, Boston (2010)Google Scholar
  12. 12.
    Fournais S., Persson M.: Strong diamagnetism for the ball in three dimensions. Asymptot. Anal. 72(1-2), 77–123 (2011)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Fournais, S., Persson Sundqvist, M.: A uniqueness theorem for higher order anharmonic oscillators. J. Spectr. Theory (2014)Google Scholar
  14. 14.
    Giorgi T.: Superconductors surrounded by normal materials. Proc. R. Soc. Edinb. Sect. A 135(2), 331–356 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Giorgi, T., Phillips, D.: The breakdown of superconductivity due to strong fields for the Ginzburg–Landau model. SIAM J. Math. Anal. 30(2), 341–359 (electronic) (1999)Google Scholar
  16. 16.
    Giorgi T., Smits R.G.: Remarks on the existence of global minimizers for the Ginzburg–Landau energy functional. Nonlinear Anal. 53(2), 147–155 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Helffer B., Mohamed A.: Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique. Ann. Inst. Fourier (Grenoble) 38(2), 95–112 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kachmar A., Persson M.: On the essential spectrum of magnetic Schrödinger operators in exterior domains. Arab J. Math. Sci. 19(2), 217–222 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Little W.A., Parks R.D.: Observation of quantum periodicity in the transition temperature of a superconducting cylinder. Phys. Rev. Lett. 9(1), 9–12 (1962)CrossRefADSGoogle Scholar
  20. 20.
    Lu K., Pan X.-B.: Estimates of the upper critical field for the Ginzburg–Landau equations of superconductivity. Phys. D 127(1–2), 73–104 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Raymond N.: Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2. Ann. Henri Poincaré 10(1), 95–122 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  22. 22.
    Rubinstein J., Schatzman M.: Asymptotics for thin superconducting rings. J. Math. Pures Appl. (9) 77(8), 801–820 (1998)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Saint-James D.: Etude du champ critique H c3 dans une géométrie cylindrique. Phys. Lett. 15(1), 13–15 (1965)CrossRefADSGoogle Scholar
  24. 24.
    Sauvageot M.: Classification of symmetric vortices for the Ginzburg–Landau equation. Differ. Integral Equ. 19(7), 721–760 (2006)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsAarhus UniversityAarhusDenmark
  2. 2.Department of Mathematical SciencesLund UniversityLundSweden

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