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

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 masks 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. The proper handling of Pauli blocking between a fixed number of composite bosons is however known to be quite difficult. To do it, we here develop a commutator formalism for Cooper pair condensate, along the line we used for excitons. We then rederive, within the N-pair subspace, a few results of BCS superconductivity obtained in the grand canonical ensemble, to evidence their Pauli blocking origin. We end by reconsidering what should be called “Cooper pair wave function” and conclude differently from usual understanding.