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Progress of Theoretical Physics

A New Boson Realization of the Two-Level Pairing Model in a Many-Fermion System and Its Classical Counterpart: -- The Role of the su(2) su(1,1)-Coherent State in the Schwinger Boson Representation for the su(2) su(2)-Algebra --

Constança Providência, João da Providência, Y. Tsue, M. Yamamura

Published Apr 1, 2006• 0 citations
Publisher

Abstract

A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.

DOI
10.1143/ptp.115.739

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A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.
A new method for describing the two-level pairing model in a many-fermion system is presented. Basic idea comes from the Schwinger boson representation for the su(2) cross su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the mixed-mode coherent state is mainly discussed in this paper.
Various classical counterparts to the two-level pairing model in a many-fermion system are presented in the Schwinger boson representation. It is shown that one of the key ingredients giving the classical descriptions for quantal systems is the use of various trial states in addition to the su(2) ⊗ su(2)-coherent state, which may be a natural selection for the two-level pairing model governed by the su(2) ⊗ su(2)-algebra. It is pointed out that a spurious behavior such as a sharp phase transition can be avoided by using the other states including the su(2) ⊗ su(1,1)- and su(1,1) ⊗ su(1,1)-coherent states. This contrasts with the situation regarding the usual Hartree-Fock-Bogoliubov and quasi-particle random phase approximations applied to the original fermion system, which yield a sharp phase transition.
Following an idea, recently, proposed by the present authors for the two-level pairing model, the Lipkin model is reexamined. It is a natural generalization of the method already developed by the present authors with Kuriyama. This model is a schematic model for many-fermion systems and obeys the su(2)-algebra. It is shown that the use of the Schwinger boson representation, the model is expressed in terms of the su(1,1) ⊗ su(1,1)-algebra and with the aid of the MYT mapping method, it is disguised from the orginal form in terms of the Holstein-Primakoff representation. Further, under various coherent states, the classical counterparts are derived. It is concluded that the Lipkin model can be treated in the common ring as that of the two-level pairing model.
Considering the present situation of Lie-algebraic studies of high-temperature superconductivity, a Schwinger-type boson realization of the su(4) algebra is investigated. The basic framework comes from a possible form of the su(n) and the su(n,1) algebras in the Schwinger representation, which was recently proposed by two of the present authors (M. Y. & A. K.), together with J. da Providência. The form investigated in this paper is, in some sense, a natural extension of the Schwinger boson representation for the so(5), so(4) and su(2)⊗su(2) algebras, which was also proposed by the present authors. The orthogonal set for the irreducible representation of the su(4) algebra is completely analogous to that expressed in terms of fermion operators for pairing correlation.