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International Journal of Modern Physics E

MEAN-FIELD PLUS VARIOUS TYPES OF PAIRING INTERACTIONS AND AN EXACT BOSON MAPPING OF THE REDUCED BCS PAIRING INTERACTION

Feng Pan, Mingxia Xie, HONG CHEN, WEI BA, QUAN YUAN, Jerry Paul Draayer

Published Dec 1, 2008• 0 citations
Publisher

Abstract

Exact solutions of Nilsson mean-field plus various type of pairing interactions are briefly reviewed. Some even-odd mass differences and moment of inertia of low-lying states for rare earth and actinide nuclei calculated in nearest-orbit pairing and extended pairing models and comparison with the corresponding experimental data are shown. An exact boson mapping of the reduced BCS pairing Hamiltonian is reported. In the mapping, fermion pair operators are mapped exactly to the corresponding bosons. The image of the mapping results in a Bose-Hubbard model with level dependent hopping.

DOI
10.1142/s0218301308012002

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Exact solutions of Nilsson mean-field plus various type of pairing interactions are briefly reviewed. Some even-odd mass differences and moment of inertia of low-lying states for rare earth and actinide nuclei calculated in nearest-orbit pairing and extended pairing models and comparison with the corresponding experimental data are shown. An exact boson mapping of the reduced BCS pairing Hamiltonian is reported. In the mapping, fermion pair operators are mapped exactly to the corresponding bosons. The image of the mapping results in a Bose-Hubbard model with level dependent hopping.
An exact boson mapping of the deformed mean-field plus equal strength pairing Hamiltonian is considered. In the mapping, fermion pair operators are mapped exactly to the corresponding bosons. The mapping occurs at the level of the Richardson-Gaudin equations. The image of the mapping results is a Bose-Hubbard model with level-dependent hopping. Although the resultant Bose-Hubbard Hamiltonian is non-Hermitian, a part of spectrum of the Bose-Hubbard Hamiltonian with U/t = 1 determined by the corresponding Richardson-Gaudin (Bethe ansatz) equations corresponds exactly to the whole spectrum of the pairing Hamiltonian. © 2009 IOP Publishing Ltd.
An exact boson mapping of the reduced BCS (equal strength) pairing Hamiltonian is considered. In the mapping, fermion pair operators are mapped exactly to the corresponding bosons. The image of the mapping results in a Bose-Hubbard model with level dependent hopping. Though the resultant Bose-Hubbard Hamiltonian is non-Hermitian, all eigenvalues are real when Uk/t< 1, where k is the total number of bosons. When U/t=1, a part of spectrum of the Bose-Hubbard Hamiltonian corresponds exactly to the whole spectrum of the reduced BCS pairing Hamiltonian.
A new step-by-step diagonalization procedure for evaluating exact solutions of the nuclear deformed mean-field plus pairing interaction model is proposed via a simple Bethe ansatz in each step from which the eigenvalues and corresponding eigenstates can be obtained progressively. This new approach draws upon an observation that the original one- plus two-body problem in a kkk-particle Hilbert subspace can be mapped onto a one-body grand hard-core boson picture that can be solved step by step with a simple Bethe ansatz known from earlier work. Based on this new procedure, it is further shown that the extended pairing model for deformed nuclei [Feng Pan, V. G. Gueorguiev, and J. P. Draayer, Phys. Rev. Lett. 92, 112503 (2004)] is similar to the standard pairing model with the first step approximation, in which only the lowest energy eigenstate of the standard pure pairing interaction part is taken into consideration. Our analysis shows that the standard pairing model with the first step approximation displays similar pair structures of the first few exact low-lying states of the model, which, therefore, provides a link between the two models.
The validity of the essential ingredients of the microscopic formulation of the interacting boson model: (i) truncation of the shell-model or fermion space and (ii) the procedures for mapping the truncated fermion space on to the boson space, is tested over various (spherical, deformed and transitional) regions both in the simplistic single j-shell, identical boson systems and the more realistic multi j-shell, proton-neutron systems depending on the feasibility of such calculations. In the situations where the truncation and mapping procedures fail to reproduce the exact shell-model results, it is demonstrated that renormalization and higher order mapping procedures cure the associated problems, respectively and provide excellent agreements. In addition, a Hamiltonian of the interacting boson-fermion model which is relevant for a study of odd mass nuclei is derived for a single j-shell case in the shell-model framework, which yields results identical to those in the shell-model for the monopole pairing interaction.
Variational calculations within the pairing plus quadrupole model are carried out for a set of particles moving in a single j -shell and for values of the coupling constants giving rise to the main features of strongly deformed nuclei. Two calculations are carried out and the corresponding results compared. In the first, particles are allowed to align in the mean field (particle aligned model). In the second, pairs of particles coupled to angular momentum λ = 0 and λ = 2 align themselves in the average field (pair aligned model). Although many features of the results are qualitatively similar, even for well deformed systems, there are important quantitative differences. In particular, the single-particle occupancies predicted by the pair aligned model display a diffusivity around the Fermi surface (effective pairing gap) which is ∼ 1.7 times larger than that predicted by the particle aligned model. The quadrupole moment predicted by the pair aligned model at maximum deformation is ∼ 1.4 smaller than that predicted by the particle aligned model. The pair aligned model and the particle aligned model thus describe two many-body systems which are rather different. In the first pairing correlations are as important as quadrupole correlations while in the second the quadrupole deformation plays the dominant role. The effect of allowing pairs of particles to couple to angular momenta λ > 2 is found to be essential to achieve quantitative agreement between the two models.