Microscopic derivation of Ginzburg-Landau theory
HTML articles powered by AMS MathViewer
- by Rupert L. Frank, Christian Hainzl, Robert Seiringer and Jan Philip Solovej
- J. Amer. Math. Soc. 25 (2012), 667-713
- DOI: https://doi.org/10.1090/S0894-0347-2012-00735-8
- Published electronically: March 26, 2012
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
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. (2) 108 (1957), 1175–1204. MR 95694
- William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556
- Søren Fournais and Bernard Helffer, Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, vol. 77, Birkhäuser Boston, Inc., Boston, MA, 2010. MR 2662319
- Rupert L. Frank, Christian Hainzl, Serguei Naboko, and Robert Seiringer, The critical temperature for the BCS equation at weak coupling, J. Geom. Anal. 17 (2007), no. 4, 559–567. MR 2365659, DOI 10.1007/BF02937429
- P.G. de Gennes, Superconductivity of Metals and Alloys, Westview Press (1966).
- V.L. Ginzburg, L.D. Landau, On the theory of superconductivity, Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950).
- 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).
- S. Gustafson, I. M. Sigal, and T. Tzaneteas, Statics and dynamics of magnetic vortices and of Nielsen-Olesen (Nambu) strings, J. Math. Phys. 51 (2010), no. 1, 015217, 16. MR 2605850, DOI 10.1063/1.3280039
- Christian Hainzl, Eman Hamza, Robert Seiringer, and Jan Philip Solovej, The BCS functional for general pair interactions, Comm. Math. Phys. 281 (2008), no. 2, 349–367. MR 2410898, DOI 10.1007/s00220-008-0489-2
- Christian Hainzl, Mathieu Lewin, and Robert Seiringer, A nonlinear model for relativistic electrons at positive temperature, Rev. Math. Phys. 20 (2008), no. 10, 1283–1307. MR 2466817, DOI 10.1142/S0129055X08003547
- Christian Hainzl, Mathieu Lewin, and Éric Séré, Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation, Comm. Math. Phys. 257 (2005), no. 3, 515–562. MR 2164942, DOI 10.1007/s00220-005-1343-4
- C. Hainzl, R. Seiringer, Critical temperature and energy gap in the BCS equation, Phys. Rev. B 77, 184517 (2008).
- Christian Hainzl and Robert Seiringer, The BCS critical temperature for potentials with negative scattering length, Lett. Math. Phys. 84 (2008), no. 2-3, 99–107. MR 2415542, DOI 10.1007/s11005-008-0242-y
- Christian Hainzl and Robert Seiringer, Spectral properties of the BCS gap equation of superfluidity, Mathematical results in quantum mechanics, World Sci. Publ., Hackensack, NJ, 2008, pp. 117–136. MR 2466682, DOI 10.1142/9789812832382_{0}009
- 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).
- B. Helffer and D. Robert, Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal. 53 (1983), no. 3, 246–268 (French, with English summary). MR 724029, DOI 10.1016/0022-1236(83)90034-4
- Andrzej Pękalski and Jerzy A. Przystawa (eds.), Modern trends in the theory of condensed matter, Lecture Notes in Physics, vol. 115, Springer-Verlag, Berlin-New York, 1980. MR 583566
- A.J. Leggett, Quantum Liquids, Oxford (2006).
- Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
- Elliott H. Lieb and Robert Seiringer, The stability of matter in quantum mechanics, Cambridge University Press, Cambridge, 2010. MR 2583992
- Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
- Didier Robert, Autour de l’approximation semi-classique, Progress in Mathematics, vol. 68, Birkhäuser Boston, Inc., Boston, MA, 1987 (French). MR 897108
- Etienne Sandier and Sylvia Serfaty, Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications, vol. 70, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2279839
- Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
- Walter Thirring, Quantum mathematical physics, 2nd ed., Springer-Verlag, Berlin, 2002. Atoms, molecules and large systems; Translated from the 1979 and 1980 German originals by Evans M. Harrell II. MR 2133871, DOI 10.1007/978-3-662-05008-8
Bibliographic Information
- Rupert L. Frank
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 728268
- ORCID: 0000-0001-7973-4688
- Email: rlfrank@math.princeton.edu
- Christian Hainzl
- Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
- Email: christian.hainzl@uni-tuebingen.de
- Robert Seiringer
- Affiliation: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Canada
- Email: robert.seiringer@mcgill.ca
- Jan Philip Solovej
- Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
- ORCID: 0000-0002-0244-1497
- Email: solovej@math.ku.dk
- 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 - © Copyright 2012 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
- Journal: J. Amer. Math. Soc. 25 (2012), 667-713
- MSC (2010): Primary 35A15, 81Q20, 82D50, 82D55, 35Q56
- DOI: https://doi.org/10.1090/S0894-0347-2012-00735-8
- MathSciNet review: 2904570