Eur. Phys. J. B 79, 263-274 (2011)
https://doi.org/10.1140/epjb/e2010-10560-7

Coboson formalism for Cooper pairs and its application to Richardson's equations

¹ Institut des NanoSciences de Paris, Universite Pierre et Marie Curie, CNRS, Tour 22, 4 place Jussieu, 75005 Paris, France
² Department of Physics, University of Illinois at Urbana-Champaign, 1110 W Green St, Urbana, IL, 61801, USA

Corresponding author: ^a monique.combescot@insp.jussieu.fr

Received: 21 July 2010
Revised: 30 November 2010
Published online: 19 January 2011

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 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.