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Register a new userA Possible Boson Realization of Generalized Lipkin Model for Many-Fermion System
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Authors
A.Kuriyama
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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.
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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.
A possible form of the Lipkin model obeying the su(6)-algebra is presented. It is a natural generalization from the idea for the su(4)-algebra recently proposed by the present authors. All the relation appearing in the present form can be expressed in terms of the spherical tensors in the su(2)-algebras. For specifying the linearly independent basis completely, twenty parameters are introduced. It is concluded that, in these parameters, the ten denote the quantum numbers coming from the eigenvalues of some hermitian operators. The five in these ten determine the minimum weight state.
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.
In this work we present an extended version of the Friedrichs Model, which includes fermion-boson couplings. The set of fermion bound states is coupled to a boson field with discrete and continuous components. As a result of the coupling some of the fermion states may become resonant states. This feature suggests the existence of a formal link between the occurrence of Gamow Resonant States in the boson sector, as predicted by the standard Friedrichs Model, with similar effects in the set of solutions of the fermion central potential (Gamow fermion resonances). The structure of the solutions of the model is discussed by using different approximations to the model space. Realistic couplings constants are used to calculate fermion resonances in a heavy mass nucleus.
With the aim of performing an argument supplement to the previous paper by the present authors, 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 on 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 the points (i) and (ii) is simple.
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