From one to $N$ Cooper pairs, step by step

TL;DR

This paper analytically extends the Cooper pair model to many pairs, deriving the BCS condensation energy and revealing a new interpretation of the pairing energy reduction due to Pauli exclusion.

Contribution

It provides an analytical solution for multiple Cooper pairs and offers a novel understanding of the BCS condensation energy based on pair interactions and Pauli blocking effects.

Findings

Exact expression for N-pair energy in dilute limit

Extension of first-order results to dense regime matches BCS energy

Proposes a new interpretation of the BCS condensation energy

Abstract

We extend the one-pair Cooper configuration towards Bardeen-Cooper-Schrieffer (BCS) model of superconductivity by adding one-by-one electron pairs to an energy layer where a small attraction acts. To do it, we solve Richardson's equations analytically in the dilute limit of pairs on the one-Cooper pair scale. We find, through only keeping the first order term in this expansion, that the $N$ correlated pair energy reads as the energy of $N$ isolated pairs within a $N(N-1)$ correction induced by the Pauli exclusion principle which tends to decrease the average pair binding energy when the pair number increases. Quite remarkably, extension of this first-order result to the dense regime gives the BCS condensation energy exactly. This leads us to suggest a different understanding of the BCS condensation energy with a pair number equal to the number of pairs feeling the potential and an average pair binding energy reduced by Pauli blocking to half the single Cooper pair energy - instead of the more standard but far larger superconducting gap.