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Register a new userNew boson realization of the Lipkin model obeying the su(2)-algebra
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New boson representation of the su(2)-algebra proposed by the present authors for describing the damped and amplified oscillator is examined in the Lipkin model as one of simple many-fermion models. This boson representation is expressed in terms of two kinds of bosons with a certain positive parameter. In order to describe the case of any fermion number, third boson is introduced. Through this examination, it is concluded that this representation is well workable for the boson realization of the Lipkin model in any fermion number.
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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 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. The cases n=2, 3, 4 and 5 are discussed in rather detail.
With the use of two kinds of boson operators, a new boson representation of the su(2)-algebra is proposed. The basic idea comes from the pseudo su(1,1)-algebra recently given by the present authors. It forms a striking contrast to the Schwinger boson representation of the su(2)-algebra which is also based on two kinds of bosons. This representation may be suitable for describing time-dependence of the system interacting with the external environment in the framework of the thermo field dynamics formalism, i.e., the phase space doubling. Further, several deformations related to the su(2)-algebra in this boson representation are discussed. On the basis of these deformed algebra, various types of time-evolution of a simple boson system are investigated.
On the basis of the formalism proposed by three of the present authors (A.K., J.P.and M.Y.), generalized Lipkin model consisting of (M+1) single-particle levels is investigated. This model is essentially a kind of the su(M+1)-algebraic model and, in contrast to the conventional treatment, the case, where fermions are partially occupied in each level, is discussed. The scheme for obtaining the orthogonal set for the irreducible representation is presented.
We study the proton-neutron RPA with an extended Lipikin-Meshkov-Glick model. We pay attention to the effect of correlated ground state and the case in which neutron and proton numbers are different. The effect of the correlated ground state are tested on the basis of quasi-boson approximation. We obtain the result that RPA excitation energies and transition strengths are in a good agreement with the exact solution up to a certain strength of the particle-particle interaction. However, the transition strength becomes worse if we consider the case in which neutron and proton numbers are different even at a weak particle-particle interaction.
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