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arXiv:nucl-th/0610074 · Progress of Theoretical Physics

Classical and Quantal Descriptions of Small Amplitude Fluctuations around Equilibriums in the Two-Level Pairing Model

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

Published May 1, 2007• 0 citations
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Abstract

Various classical counterparts to the two-level pairing model in a many-fermion system are presented in the Schwinger boson representation. It is shown that one of the key ingredients giving the classical descriptions for quantal systems is the use of various trial states in addition to the su(2) ⊗ su(2)-coherent state, which may be a natural selection for the two-level pairing model governed by the su(2) ⊗ su(2)-algebra. It is pointed out that a spurious behavior such as a sharp phase transition can be avoided by using the other states including the su(2) ⊗ su(1,1)- and su(1,1) ⊗ su(1,1)-coherent states. This contrasts with the situation regarding the usual Hartree-Fock-Bogoliubov and quasi-particle random phase approximations applied to the original fermion system, which yield a sharp phase transition.

DOI
10.1143/ptp.117.835

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Various classical counterparts to the two-level pairing model in a many-fermion system are presented in the Schwinger boson representation. It is shown that one of the key ingredients giving the classical descriptions for quantal systems is the use of various trial states in addition to the su(2) ⊗ su(2)-coherent state, which may be a natural selection for the two-level pairing model governed by the su(2) ⊗ su(2)-algebra. It is pointed out that a spurious behavior such as a sharp phase transition can be avoided by using the other states including the su(2) ⊗ su(1,1)- and su(1,1) ⊗ su(1,1)-coherent states. This contrasts with the situation regarding the usual Hartree-Fock-Bogoliubov and quasi-particle random phase approximations applied to the original fermion system, which yield a sharp phase transition.
A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.
A new method for describing the two-level pairing model in a many-fermion system is presented. Basic idea comes from the Schwinger boson representation for the su(2) cross su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the mixed-mode coherent state is mainly discussed in this paper.
The two-level pairing model was investigated in the framework of the Schwinger representation. It has been shown that zero-point fluctuations, which are important in the crossover region, are well described by certain coherent states which conserve the constants of motion. These so-called conserving approximations provide excellent descriptions of the ground-state energy and of the first excitation energy over the whole range of coupling constant values. The description of the ground-state energy provided by the Glauber coherent state is also reasonable, but not so good. Although the Glauber coherent state fails to provide the description of the excitation energy around the critical point, its performance in the strong coupling limit is very good both for the description of the ground-state energy and of the first excitation energy. The present results also show that already at the mean-field level, the bosonic description in the independent particle approximation incorporates important correlation corrections to the conventional mean-field description based on the fermionic realization of the su (2) algebra (BCS theory).
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.