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Journal of Mathematical Physics

Boson Expansions for an Exactly Soluble Model of Interacting Fermions with SU(3) Symmetry

Shou Yung Li, Abraham Klein, R. M. Dreizler

Published Mar 1, 1970• 81 citations
Publisher

Abstract

An exactly soluble three-level model for a system of fermions with Hamiltonian formed from generators of the group SU(3) is studied. A basis for the representation of which the ground state is a member is constructed. It is demonstrated that in this representation, the generators can each be expanded in a series as functions of a pair of ``kinematical'' boson operators; the series are uniquely determined to satisfy the operator algebra and the invariants of the representation by a method of Marumori. It is seen that the lowest anharmonic approximation to the Hamiltonian and other operators yields excellent numerical agreement with exact results for all regimes of interaction strength considered. An alternative description of the system in terms of a dynamically more meaningful boson, called the ``physical'' boson, is shown to be appropriate for relatively weak coupling where one has a near harmonic spectrum.

DOI
10.1063/1.1665234

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A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
An algorithm for obtaining a recursive algebraic expression for the various expansion coefficients appearing in the physical boson basis states is presented, using the Dyson boson mapping. It is observed that only the first few terms of the physical boson basis states, expressed in the present form, contribute to the matrix element of the Hamiltonian. Explicit numerical calculations have been performed for Ni and Sn isotopes in the obtained physical boson basis states. The pairing Hamiltonian has been used as the model interaction. The results are more than 90% in agreement with exact shell model calculations.
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A general class of exactly soluble three-level models is presented for an NNN-fermion system. Assuming a monopole-monopole interaction among the fermions allows one to express the model Hamiltonian in terms of the generators of the SU3S{U}_{3}SU3​ group. A specialized version of this class of models is adapted for the study of mixing two collective states of opposite parity, analogous to the mixing of giant-dipole-resonance and surface quadrupole vibrations in nuclei. Exact results of the model are studied as a function of interaction parameters and compared with approximations including up to three-particle-three-hole excitations. It is shown that in the model many-particle-many-hole states play an important role in the mixing of collective states and cannot be properly neglected. The implications of this result for the study of actual nuclei is discussed.
The Beliaev-Zelivinsky boson expansion and particle-vibration coupling methods are studied in comparison with an exact treatment of the monopole model. Two of the three boson expansions considered were valid for all interaction strengths. A third expansion and the particle-vibration coupling method were found to be equivalent to perturbation theory. The difficulty associated with the overcompleteness of the basis in the particle-vibration coupling method was resolved both in the computation of the energies and the matrix element of the monopole specific operator.