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arXiv:0707.3787 · Theoretical and Mathematical Physics

Effective su q(2) Models and Polynomial Algebras for Fermion–Boson Hamiltonians

Ángel Ballesteros, Osvaldo Civitarese, Francisco J. Herranz, M. Reboiro

Published Nov 1, 2003• 6 citations
arXivHTMLPDF
DOI
10.1023/a:1027301616731

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A pseudo su(1,1)-algebra is formulated as a possible deformation of the Cooper pair in the su(2)-algebraic many-fermion system. With the aid of this algebra, it is possible to describe the behavior of individual fermions which are generated as the result of interaction with the external environment. The form presented in this paper is a generalization of a certain simple case developed recently by the authors. The basic idea follows the su(1,1) algebra in the Schwinger boson representation for treating energy transfer between the harmonic oscillator and the external environment. The Hamiltonian is given following the idea of phase space doubling in the thermo-field dynamics formalism, and the time-dependent variational method is applied to this Hamiltonian. Its trial state is constructed in the frame deformed from the BCS-Bogoliubov approach to superconductivity. Several numerical results are shown.
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.
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.