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arXiv:1007.1302

Two-Level Lipkin Model in Unconventional Boson Realization

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

Published Jul 8, 2010• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1007.1302

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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.
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.
Following the Schwinger boson representation for the su(M + 1)- and su(N, 1)-algebras presented by two of the present authors (J. da P. and M. Y.) and Kuriyama, a possible counterpart of the Lipkin model in the su(M + 1)-algebra formulated in the fermion space is presented. The free vacuum, which plays a fundamental role in the conventional treatment of the Lipkin model, is generalized in a quite natural way, and further, the excited state generating operators, such as the particle-hole pairs, are also given in a natural scheme. As concrete examples, the cases of the su(2)-, su(3)- and su(4)-algebras are studied. Especially, the case of the su(4)-algebra is investigated in detail in relation to the nucleon pairing correlations and the high-temperature superconductivity.
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.