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arXiv:1709.06699 · Progress of Theoretical and Experimental Physics

A practical scheme for constructing the minimum-weight states of the su(n)-Lipkin model in arbitrary fermion number

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

Published Oct 25, 2017• 0 citations
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Abstract

With the aim of performing an argument supplementary to the previous paper by the present authors [Y. Tsue, C. Providência, J. da Providência, and M. Yamamura, Prog. Theor. Exp. Phys. 2017, 081D01 (2017)], in this paper, a practical scheme for constructing the minimum-weight states of the |$su(n)$|-Lipkin model in arbitrary fermion number is discussed. The idea comes from the following two points: (i) consideration of the property of one-fermion transfer induced by the |$su(n)$|-generators in the Lipkin model and (ii) use of the auxiliary |$su(2)$|-algebra presented by the present authors. The form obtained under points (i) and (ii) is simple.

DOI
10.1093/ptep/ptx158

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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...............
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 been already 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 following this idea, the minimum weight states are determined for any fermion number occupying appropriately 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.
Standing on the results for the minimum weight states obtained in the previous paper (I), an idea how to construct the linearly independent basis is proposed for the su(n)-Lipkin model. This idea starts in setting up m independent su(2)-subalgebras in the cases with n=2m and n=2m+1 (m=2,3,4,...). The original representation is re-formed in terms of the spherical tensors for the su(n)-generators built under the su(2)-subalgebras. Through this re-formation, the su(m)-subalgebra can be found. For constructing the linearly independent basis, not only the su(2)-algebras but also the su(m)-subalgebra play a central role. Some concrete results in the cases with n=2, 3, 4 and 5 are presented.
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.