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Progress of Theoretical Physics

The Lipkin Model in a New Boson Realization: Basic Idea

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

Published Apr 1, 2000• 3 citations
Publisher

Abstract

With the use of the basic idea of the boson mapping method, a new boson realization, which is suitable for the Lipkin model, is proposed. This realization is different from the forms which are obtained with the Holstein-Primakoff and the Schwinger boson representations for the su(2) spin system. Further, with the aid of the basic idea of the TDHF method in canonical form, a method analysing the Lipkin model in the framework of classical mechanics is developed. The condition for the solution of the Hamilton equation of motion is discussed.

DOI
10.1143/ptp.103.733

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With the use of the basic idea of the boson mapping method, a new boson realization, which is suitable for the Lipkin model, is proposed. This realization is different from the forms which are obtained with the Holstein-Primakoff and the Schwinger boson representations for the su(2) spin system. Further, with the aid of the basic idea of the TDHF method in canonical form, a method analysing the Lipkin model in the framework of classical mechanics is developed. The condition for the solution of the Hamilton equation of motion is discussed.
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.
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.
The su(2)-algebraic many-fermion model is formulated so as to be able to get the unified understanding of the structures of three simple models: the single-level pairing, the isoscalar proton-neutron pairing and the two-level Lipkin model. The basic idea is to introduce an auxiliary su(2)-algebra, any generator of which commutes with any generator of the starting su(2)-algebra. With the aid of this algebra, the minimum weight states are completely determined in simple form. Further, concerning the two algebras, boson realization is presented. Through this formulation, the behavior of the total fermion in the Lipkin model is notably different from that in the other two models. As the supplementary problem, the boson-fermion realization and the Lipkin model in the isovector pairing model are investigated.