Coboson formalism for Cooper pairs used to derive Richardson's equations

TL;DR

This paper introduces a many-body formalism for Cooper pairs, using Shiva diagrams, to derive Richardson's equations and analyze the ground state energy dependence, providing new insights into the BCS wave function and dense regimes.

Contribution

It develops a coboson formalism for Cooper pairs, deriving Richardson's equations and analyzing the energy dependence in dense regimes, offering new understanding of many-body effects.

Findings

N Cooper pairs differ from single pairs only by electron exchange.

The N(N-1) dependence of the ground state energy remains valid in dense regimes.

The BCS wave function ansatz is questioned compared to Richardson's exact solution.

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 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 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 even 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.