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Coupling Schemes for an n su(2) Spin System

Masatoshi Yamamura, Constança Providência, João Da Providência, Yasuhiko Tsue, João Da Providência

Published Jan 1, 2004• 0 citations
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Abstract

In the framework of the Schwinger boson representation for the su(2)-algebra, the closed form is derived for the total spin eigenstates which result from the coupling of n su(2)-spins. In order to demonstrate its usefulness, the orthogonal set for the so(5)-algebra, which is reduced to four su(2)-spin systems, is obtained. 1 typeset using PTPTEX.sty <ver.1.0>§1. Introduction and preliminaries The boson realization of the su(2)-algebra due to Schwinger 1) and named after him plays a crucial role in the Theory of Angular Momentum, for instance in the derivation of the irreducible representations of the rotation operators. The Schwinger representation of the su(2)-algebra also proved to be most useful in the study of various many-body systems,

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In the framework of the Schwinger boson representation for the su(2)-algebra, the closed form is derived for the total spin eigenstates which result from the coupling of n su(2)-spins. In order to demonstrate its usefulness, the orthogonal set for the so(5)-algebra, which is reduced to four su(2)-spin systems, is obtained.

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In the framework of the Schwinger boson representation for the su(2)-algebra, the closed form is derived for the total spin eigenstates which result from the coupling of n su(2)-spins. In order to demonstrate its usefulness, the orthogonal set for the so(5)-algebra, which is reduced to four su(2)-spin systems, is obtained. 1 typeset using PTPTEX.sty <ver.1.0>§1. Introduction and preliminaries The boson realization of the su(2)-algebra due to Schwinger 1) and named after him plays a crucial role in the Theory of Angular Momentum, for instance in the derivation of the irreducible representations of the rotation operators. The Schwinger representation of the su(2)-algebra also proved to be most useful in the study of various many-body systems,
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.
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.
With the use of two kinds of boson operators, a new boson representation of the |su(2)su(2)su(2)|-algebra is proposed. The basic idea comes from the pseudo |su(1,1)su(1,1)su(1,1)|-algebra recently given by the present authors [Y. Tsue et al., Prog. Theor. Exp. Phys. 2013, 103D04 (2013)]. It forms a striking contrast to the Schwinger boson representation of the |su(2)su(2)su(2)|-algebra, which is also based on two kinds of bosons. It is proved that this new boson representation obeys the |su(2)su(2)su(2)|-algebra in a certain subspace in the whole boson space constructed by the Schwinger boson representation of the |su(1,1)su(1,1)su(1,1)|-algebra. This representation may be suitable for describing the time dependence of the system interacting with the external environment in the framework of the thermo field dynamics formalism, i.e., phase space doubling. Further, several deformations related to the |su(2)su(2)su(2)|-algebra in this boson representation are discussed. On the basis of these deformed algebras, various types of time evolution of a simple boson system are investigated.