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Path integral bosonization of the pairing Hamiltonian

M. B. Barbaro, Alfredo Molinari, F. Palumbo, M. Quaglia

Published Apr 9, 2003• 0 citations
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Abstract

As a first approach to a derivation of the IBM from a microscopic nuclear hamiltonian, we bosonize the pairing hamiltonian in the framework of the path integral formalism respecting both the particle number conservation and the Pauli principle. We construct the saddle point expansion which reproduces the sector of the spectrum associated to the addition or removal of nucleon pairs. 1

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Progress of Theoretical Physics Supplement•May 16, 2013•1 citation
As a first approach to a derivation of the IBM from a microscopic nuclear hamiltonian, we bosonize the pairing hamiltonian in the framework of the path integral formalism respecting both the particle number conservation and the Pauli principle. We construct the saddle point expansion which reproduces the sector of the spectrum associated to the addition or removal of nucleon pairs. 1
The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.
We derive a boson Hamiltonian from a Nuclear Hamiltonian whose potential is expanded in pairing multipoles and determine the fermion-boson mapping of operators. We use a new method of bosonization based on the evaluation of the partition function restricted to the bosonic composites of interest. By rewriting the partition function so obtained in functional form we get the euclidean action of the composite bosons from which we can derive the Hamiltonian. Such a procedure respects all the fermion symmetries.
An earlier work dealing with low-lying collective states in nuclei is revised and improved. Correlated and independent nucleon and pairs properly modified so that they behave approximately like bosons are constructed in terms of which a model space for low-lying collective states is defined, and a microscopic basis for IBM is thus set up. All dynamical variables can also be "bosonized", i.e., expressed in terms of these "bosons" but with state-dependent coefficients or operators to incorporate their state dependence in model space.
A pairing model for nucleons, introduced by Richardson in 1966, which describes proton–neutron pairing as well as proton–proton and neutron–neutron pairing, is re-examined in the context of the quantum inverse scattering method. Specifically, this shows that the model is integrable by enabling the explicit construction of the conserved operators. We determine the eigenvalues of these operators in terms of the Bethe ansatz, which in turn leads to an expression for the energy eigenvalues of the Hamiltonian.
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.