The BCS-Bogoliubov and the su(2)-Algebraic Approach to the Pairing Model in Many-Fermion System

TL;DR

This paper explores the relationship between the BCS-Bogoliubov and algebraic su(2) approaches to the many-fermion pairing model, demonstrating their equivalence and proposing a modification of the BCS method.

Contribution

It clarifies the connection between two fundamental approaches to the fermion pairing model and introduces a modified BCS-Bogoliubov approach based on quasiparticles with conserved fermion number.

Findings

The two approaches are shown to be equivalent via c-number replacement.

Quasiparticles in the paper reduce to those in the BCS-Bogoliubov approach under certain conditions.

A modified BCS-Bogoliubov approach is proposed and discussed.

Abstract

The relation between two approaches to the su(2)-algebraic many-fermion model is discussed: (1) the BCS-Bogoliubov approach in terms of the use of the quasiparticles representing all the degrees of freedom except those related to the Cooper-pairs and (2) the conventional algebraic approach in terms of the use of the minimum weight states, from which the Cooper-pairs are excluded. In order to arrive at the goal, the idea of the quasiparticles is brought up in the conservation of the fermion number. Under the c-number replacement for the three su(2)-generators, the quasiparticles suggested in this paper are reduced to those in the BCS-Bogoliubov approach. It is also shown that the two approaches are equivalent through the c-number replacement. Further, a certain modification of the BCS-Bogoliubov approach is discussed.