Abstract
We propose a many-body formalism for Cooper pairs which has similarities to the one we recently developed for composite boson excitons (coboson in short). Its Shiva diagram representation evidences that N Cooper pairs differ from N single pairs through electron exchange only: no direct coupling exists due to the very peculiar form of the reduced BCS potential. As a first application, we here use this formalism to derive Richardson’s equations for the exact eigenstates of N Cooper pairs. This derivation gives hints on why the \(N(N-1)\) dependence of the N-pair ground state energy we recently obtained by solving Richardson’s equations analytically in the low density limit, stays valid up to the dense regime. No higher order dependence exists under large overlap, a surprising result hard to accept at first. We also briefly question the BCS wave function ansatz compared to Richardson’s exact form, in the light of our understanding of coboson many-body effects.
Similar content being viewed by others
References
J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 106, 162 (1957)
R.W. Richardson, Phys. Lett. 3, 277 (1963)
R.W. Richardson, N. Sherman, Nucl. Phys. 52, 221 (1964)
R.W. Richardson, J. Math. Phys. 9, 1327 (1968)
R.W. Richardson, J. Math. Phys. 18, 1802 (1977)
M. Gaudin, États Propres et Valeurs Propres de l’Hamiltonien d’Appariement (Les Éditions de Physique, France, 1995)
A.G. Ushveridze, Quasi-exactly solvable models in quantum mechanics (Taylor & Francis Group LLC, 1994), ISBN 0730302666
J. Dukelsky, S. Pittel, G. Sierra, Rev. Mod. Phys. 76, 643 (2004)
G. Ortiz, R. Somma, J. Dukelsky, S. Rombouts, Nucl. Phys. B 707, 421 (2005)
F. Braun, J. von Delft, Phys. Rev. Lett. 81, 4712 (1998)
J. von Delft, D.C. Ralph, Phys. Rep. 345, 61 (2001)
G. Sierra, J. Dukelsky, G.G. Dussel, J. von Delft, F. Braun, Phys. Rev. B 61, R11890 (2000)
M. Schechter, Y. Imry, Y. Levinson, J.V. Delft, Phys. Rev. B 63, 214518 (2001)
D.M. Eagles, Phys. Rev. 186, 456 (1969)
A.J. Leggett, Modern Trends in the Theory of Condensed Matter, in Proceedings of the XVIth Karpacz Winter School of Theoretical Physics, Karpacz, Poland (Springer-Verlag, 1980), pp. 13–27
G. Ortiz, J. Dukelsky, Phys. Rev. A 72, 043611 (2005)
W.V. Pogosov, M. Combescot, JETP Lett. 92, 534 (2010)
W.V. Pogosov, M. Combescot, Submitted to Phys. Rev. B
M. Combescot, O. Betbeder-Matibet, F. Dubin, Phys. Rep. 463, 215 (2008)
M. Combescot, W.V. Pogosov, Eur. Phys. J. B 68, 161 (2009)
M. Combescot, O. Betbeder-Matibet, Eur. Phys. J. B 31, 517 (2003)
J. Bang, J. Krumlinde, Nucl. Phys. A 141, 18 (1970)
M. Hasegawa, S. Tazaki, Phys. Rev. C 35, 1508 (1987)
J.M. Román, G. Sierra, J. Dukelsky, Nucl. Phys. B 634, 483 (2002)
W.V. Pogosov, M. Combescot, M. Crouzeix, Phys. Rev. B 81, 174514 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Combescot, M., Zhu, G. Coboson formalism for Cooper pairs and its application to Richardson’s equations. Eur. Phys. J. B 79, 263–274 (2011). https://doi.org/10.1140/epjb/e2010-10560-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2010-10560-7