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Microscopic derivation of Ginzburg-Landau theory

Authors: Rupert L. Frank, Christian Hainzl, Robert Seiringer and Jan Philip Solovej
Journal: J. Amer. Math. Soc. 25 (2012), 667-713
MSC (2010): Primary 35A15, 81Q20, 82D50, 82D55, 35Q56
Published electronically: March 26, 2012
MathSciNet review: 2904570
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Abstract | References | Similar Articles | Additional Information

Abstract: We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz, I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover (1964). MR 0167642 (29:4914)
  • 2. J. Bardeen, L. Cooper, J. Schrieffer, Theory of Superconductivity, Phys. Rev. 108, 1175-1204 (1957). MR 0095694 (20:2196)
  • 3. W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer (1974). MR 0486556 (58:6279)
  • 4. S. Fournais, B. Helffer, Spectral Methods in Surface Superconductivity, Birkhäuser (2010). MR 2662319 (2011j:35003)
  • 5. R.L. Frank, C. Hainzl, S. Naboko, R. Seiringer, The critical temperature for the BCS equation at weak coupling, J. Geom. Anal. 17, 559-568 (2007). MR 2365659 (2008k:82147)
  • 6. P.G. de Gennes, Superconductivity of Metals and Alloys, Westview Press (1966).
  • 7. V.L. Ginzburg, L.D. Landau, On the theory of superconductivity, Zh. Eksp. Teor. Fiz. 20, 1064-1082 (1950).
  • 8. L.P. Gor'kov, Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity, Zh. Eksp. Teor. Fiz. 36, 1918-1923 (1959); English translation Soviet Phys. JETP 9, 1364-1367 (1959).
  • 9. S. Gustafson, I.M. Sigal, T. Tzaneteas, Statics and dynamics of magnetic vortices and of Nielsen-Olesen (Nambu) strings, J. Math. Phys. 51, 015217 (2010). MR 2605850 (2011b:82093)
  • 10. C. Hainzl, E. Hamza, R. Seiringer, J.P. Solovej, The BCS functional for general pair interactions, Commun. Math. Phys. 281, 349-367 (2008). MR 2410898 (2009d:82163)
  • 11. C. Hainzl, M. Lewin, R. Seiringer, A nonlinear theory for relativistic electrons at positive temperature, Rev. Math. Phys. 20, 1283-1307 (2008). MR 2466817 (2010g:81308)
  • 12. C. Hainzl, M. Lewin, É. Séré, Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation, Commun. Math. Phys. 257, 515-562 (2005). MR 2164942 (2006i:81123)
  • 13. C. Hainzl, R. Seiringer, Critical temperature and energy gap in the BCS equation, Phys. Rev. B 77, 184517 (2008).
  • 14. C. Hainzl, R. Seiringer, The BCS critical temperature for potentials with negative scattering length. Lett. Math. Phys. 84, 99-107 (2008). MR 2415542 (2009d:82164)
  • 15. C. Hainzl, R. Seiringer, Spectral properties of the BCS gap equation of superfluidity, in: Mathematical Results in Quantum Mechanics, proceedings of QMath10, Moeciu, Romania, September 10-15, 2007, World Scientific (2008). MR 2466682 (2010b:82073)
  • 16. C. Hainzl, R. Seiringer, Low Density Limit of BCS Theory and Bose-Einstein Condensation of Fermion Pairs, preprint arXiv:1105.1100, Lett. Math. Phys. (in press).
  • 17. B. Helffer, D. Robert, Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal. 53, 246-268 (1983). MR 724029 (85i:47052)
  • 18. A.J. Leggett, Diatomic Molecules and Cooper Pairs, in: Modern trends in the theory of condensed matter, A. Pekalski, R. Przystawa, eds., Springer (1980). MR 583566 (81h:82002)
  • 19. A.J. Leggett, Quantum Liquids, Oxford (2006).
  • 20. E.H. Lieb, M. Loss, Analysis, $ 2^{\rm nd}$ ed., Grad. Studies in Math., vol. 14, Amer. Math. Soc. (2001). MR 1817225 (2001i:00001)
  • 21. E.H. Lieb, R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge (2010). MR 2583992 (2011j:81369)
  • 22. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press (1978). MR 0493421 (58:12429c)
  • 23. D. Robert, Autour de l'approximation semi-classique, Progress in Mathematics 68, Birkhäuser (1987). MR 897108 (89g:81016)
  • 24. E. Sandier, S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Birkhäuser (2006). MR 2279839 (2008g:82149)
  • 25. B. Simon, Trace ideals and their applications, $ 2^{\rm nd}$ ed., Mathematical Surveys and Monographs, vol. 120, Amer. Math. Soc. (2005). MR 2154153 (2006f:47086)
  • 26. W. Thirring, Quantum Mathematical Physics, $ 2^{\rm nd}$ ed., Springer (2002). MR 2133871 (2006b:81368)

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Additional Information

Rupert L. Frank
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Christian Hainzl
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Robert Seiringer
Affiliation: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Canada

Jan Philip Solovej
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Received by editor(s): February 20, 2011
Received by editor(s) in revised form: November 14, 2011
Published electronically: March 26, 2012
Additional Notes: The first author gratefully acknowledges financial support received via U.S. NSF grant PHY-1068285
The second author gratefully acknowledges financial support received via U.S. NSF grant DMS-0800906
The third author gratefully acknowledges financial support received via U.S. NSF grant PHY-0845292 and NSERC
The last author gratefully acknowledges financial support received via a grant from the Danish council for independent research
Article copyright: © Copyright 2012 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

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