BCS ansatz for superconductivity in the canonical ensemble and the Pauli exclusion principle

TL;DR

This paper develops a commutator formalism for Cooper pairs in the canonical ensemble to explicitly reveal the role of Pauli exclusion in superconductivity, providing new insights into many-body effects.

Contribution

It introduces a novel commutator formalism for Cooper pairs in the canonical ensemble, highlighting the Pauli exclusion principle's influence on superconductivity.

Findings

Rederived BCS results within the N-pair subspace

Demonstrated Pauli blocking effects on Cooper pairs

Clarified the concept of 'Cooper pair wave function'

Abstract

The usual formulation of the BCS ansatz for superconductivity in the grand canonical ensemble makes the handling of the Pauli exclusion principle between paired electrons straightforward. It however tends to mask that many-body effects between Cooper pairs interacting through the reduced BCS potential are entirely controlled by this exclusion. To show it up, one has to work in the canonical ensemble. Pauli blocking between a fixed number of composite bosons is however known to be difficult to handle. To do it, we here develop a commutator formalism for Cooper pairs, along the line we used for excitons. We then rederive, within the $N$-pair subspace, a few results of BCS superconductivity commonly derived in the grand canonical ensemble, to evidence their Pauli blocking origin. We end by discussing what should be called "Cooper pair wave function".