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arXiv:nucl-th/0011068

A Possible Boson Realization of Generalized Lipkin Model for Many-Fermion System

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

Published Nov 18, 2000• 0 citations
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Abstract

On the basis of the formalism proposed by three of the present authors (A.K., J.P.and M.Y.), generalized Lipkin model consisting of (M+1) single-particle levels is investigated. This model is essentially a kind of the su(M+1)-algebraic model and, in contrast to the conventional treatment, the case, where fermions are partially occupied in each level, is discussed. The scheme for obtaining the orthogonal set for the irreducible representation is presented.

DOI
10.48550/arxiv.nucl-th/0011068

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On the basis of the formalism proposed by three of the present authors (A.K., J.P.and M.Y.), generalized Lipkin model consisting of (M+1) single-particle levels is investigated. This model is essentially a kind of the su(M+1)-algebraic model and, in contrast to the conventional treatment, the case, where fermions are partially occupied in each level, is discussed. The scheme for obtaining the orthogonal set for the irreducible representation is presented.
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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.
With the aim of applying to the Lipkin model in the case of open shell system, a possible form of the boson realization for the su(2)-algebra is proposed both in the Schwinger and the Holstein-Primakoff representation. The basic idea is borrowed from the Schwinger boson realization for the su(4)-algebra for many-quark system recently presented by the present authors. As the simplest approximation, a certain approximate form is given and the result is a natural generalization of the RPA result in the case of closed shell system.
The minimum weight states of the Lipkin model consisting of n single-particle levels and obeying the SU (n) algebra are investigated systematically.The basic idea is to use the SU (2) algebra, which is independent of the SU (n) algebra.This idea has already been presented by the present authors in the case of the conventional Lipkin model consisting of two single-particle levels and obeying the SU (2) algebra.If this idea is followed, the minimum weight states are determined for any fermion number appropriately occupying n single-particle levels.Naturally, the conventional minimum weight state is included: all fermions occupy energetically the lowest single-particle level in the absence of interaction.The cases n = 2, 3, 4, and 5 are discussed in some detail...............
A new boson representation of the |su(2)su(2)su(2)|-algebra proposed by the present authors for describing the damped and amplified oscillator is examined in the Lipkin model as one of the simple many-fermion models. This boson representation is expressed in terms of two kinds of bosons with a certain positive parameter. In order to describe the case of any fermion number, a third boson is introduced. Through this examination, it is concluded that this representation is very workable for the boson realization of the Lipkin model in any fermion number.