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 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".

BCS ansatz for superconductivity in the light of the Bogoliubov approach and the Richardson–Gaudin exact wave function

“…to possibly perform the sum over N easily, is unimportant in the thermodynamical limit, i.e., when the grand-canonical approach is valid, because the N distribution is known to be very much peaked on the N mean value -as possible to explicitly show (see Ref. [22]). The state |φ BCS then takes a compact form…”

Conditions for formation of electron pairs in a metal

“…In the part 2 of the article we consider calculations in the framework of the above-mentioned approach. As far as we know, such approach was not discussed in [1] and in the subsequent relevant research papers and reviews, including the recent works [18][19][20][21]. A mechanism of electron pairing that takes into account the results of the article is considered in parts 3 and 4; conditions for formation of electron pairs in a metal are discussed in the part 5.…”

Variational principle of Bogoliubov and generalized mean fields in many-particle interacting systems

“…In a series of papers [228,229,230], M. Combescot, W.V. Pogosov and O. Betbeder-Matibet studied various aspects of the BCS ansatz for superconductivity [190] in the light of the Bogoliubov approach.…”

“…In the next paper [229] the usual formulation of the BCS-Bogoliubov 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-Bogoliubov potential are entirely controlled by this exclusion.…”

“…Quite remarkably, extension of this first-order result to the dense regime gives the BCS-Bogoliubov condensation energy exactly. These facts may lead ones to a different interpretation of the BCS-Bogoliubov 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.In the next paper[229] the usual formulation of the BCS-Bogoliubov 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-Bogoliubov potential are entirely controlled by this exclusion.…”