Abstract
We consider the Bogolubov–Hartree–Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose–Einstein condensate of fermion pairs. The latter can be described by means of the Gross–Pitaevskii energy functional.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bach, V., Fröhlich, J., Jonsson, L.: Bogolubov-Hartree-Fock mean field theory for neutron stars and other systems with attractive interactions. J. Math. Phys 102102, 50 (2009)
Bach, V., Lieb, E.H., Solovej, J.P., Generalized Hartree-Fock theory and the Hubbard model. J. Stat Phys 76, 3–89 (1994)
Bräunlich, G., Hainzl, C., Seiringer, R.: Translation invariant quasi-free states for fermionic systems and the BCS approximation. Rev. Math. Phys 1450012, 26 (2014)
Drechsler, M., Zwerger, W.: Crossover from BCS-superconductivity to Bose-condensation. Ann. Phys 504, 15–23 (1992)
Frank, R.L., Hainzl, C., Naboko, S., Seiringer, R.: The critical temperature for the BCS equation at weak coupling. J. Geom. Anal 17, 559–567 (2007)
Frank, R.L., Hainzl, C., Seiringer, R., Solovej, J.P.: Microscopic derivation of Ginzburg-Landau theory. J. Amer. Math. Soc 25, 667–713 (2012)
Gor’kov, L.P.: Microscopic derivation of the Ginzburg–Landau equations in the theory of superconductivity. ZH. Eksp. Teor. Fiz. 36, 1918–1923 (1959)
Hainzl, C., Hamza, E., Seiringer, R., Solovej, J.P.: The BCS functional for general pair interactions. Comm. Math. Phys. 281, 349–367 (2008)
Hainzl, C., Lenzmann, E., Lewin, M., Schlein, B.: On blowup for time-dependent generalized Hartree-Fock equations. Ann. Henri Poincaré 11, 1023–1052 (2010)
Hainzl, C., Schlein, B.: Dynamics of Bose-Einstein condensates of fermion pairs in the low density limit of BCS theory. J. Funct. Anal 265, 399–423 (2013)
Hainzl, C., Seiringer, R.: Critical temperature energy gap for the BCS equation. Phys. Rev B 184517, 77 (2008)
Hainzl, C., Seiringer, R.: Low density limit of BCS theory and Bose-Einstein condensation of fermion pairs. Lett Math. Phys 100, 119–138 (2012)
Hainzl, C., Seiringer, R.: The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties. J. Math. Phys. 57, 021101 (2016)
Leggett, A.J.: Diatomic molecules and Cooper pairs. In: Modern trends in the theory of condensed matter (A. Pekalski and J. Przystawa, eds.), Lecture Notes in Physics, vol. 115, pp. 13–27, Springer Berlin / Heidelberg (1980)
Lenzmann, E., Lewin, M.: Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J 152, 257–315 (2010)
Lewin, M., Paul, S.: A numerical perspective on Hartree-Fock-Bogoliubov theory, ESAIM. Mathematical Modelling and Numerical Analysis 48, 53–86 (2014)
Noziéres, P., Schmitt-Rink, S.: Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. J. Low Temp. Phys 59, 195–211 (1985)
Pieri, P., Strinati, G.C.: Derivation of the Gross-Pitaevskii equation for condensed bosons from the Bogoliubov–de-Gennes equations for superfluid fermions. Phys Revista de Letras 030401, 91 (2003)
Randeria, M: Crossover from BCS theory to Bose-Einstein condensation. In: Bose-einstein condensation (A. Griffin, D.W. Snoke, and S. Stringari, eds.), pp. 355–392, Cambridge University Press (1996)
Sá de Melo, C.A.R., Randeria, M., Engelbrecht, J.R., Crossover from, B.C.S.: To Bose superconductivity: Transition temperature and time-dependent Ginzburg-Landau theory. Phys. Revista de Letras 71, 3202–3205 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Ⓒ2015 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bräunlich, G., Hainzl, C. & Seiringer, R. Bogolubov–Hartree–Fock Theory for Strongly Interacting Fermions in the Low Density Limit. Math Phys Anal Geom 19, 13 (2016). https://doi.org/10.1007/s11040-016-9209-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-016-9209-x
Keywords
- Bogolubov-Hartree-Fock functional
- BCS-theory
- Gross-Pitaevskii functional
- Bose-Einstein condensation
- Low density limit
- Pairing