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Journal of Physics G Nuclear and Particle Physics

Exact boson mapping of the nuclear pairing Hamiltonian

Feng Pan, Jerry Paul Draayer

Published Nov 28, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0954-3899/36/2/025103

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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.
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.
The present contribution discusses a connection between the exact Bethe Ansatz eigenstates of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and the multi-phonon states of the Tamm-Dancoff Approximation (TDA). The connection is made on the algebraic level, by means of a deformed quasi-spin algebra with a bosonic Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. Each exact Bethe Ansatz eigenstate is mapped on a unique TDA multi-phonon state, shedding light on the physics behind the Bethe Ansatz structure of the exact wave function. The procedure is illustrated with a model describing neutron pairing in 56 Fe.
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.