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AIP conference proceedings

Bosonic Treatment of Pairing Correlations in a Solvable Many-Level Model

D. Gambacurta

Published Jan 1, 2005• 0 citations
Publisher

Abstract

We discuss two extensions of RPA based on a boson formalism and report on some applications within an exactly solvable many‐level pairing model. By making use of a Marumori‐type mapping procedure, we preliminary define the boson image of the fermion operators of interest and, in particular, of the model Hamiltonian. This includes two and four‐boson terms. We therefore construct renormalized RPA‐type equations in correspondence with different expansions of the boson operators. The approximate calculations are compared with the exact ones and with those obtained within RPA and Self Consistent RPA. We find that it is possible to improve considerably upon RPA by still keeping the formalism simple enough to be applied in more realistic cases without a too serious computational effort.

DOI
10.1063/1.2140637

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We discuss two extensions of RPA based on a boson formalism and report on some applications within an exactly solvable many‐level pairing model. By making use of a Marumori‐type mapping procedure, we preliminary define the boson image of the fermion operators of interest and, in particular, of the model Hamiltonian. This includes two and four‐boson terms. We therefore construct renormalized RPA‐type equations in correspondence with different expansions of the boson operators. The approximate calculations are compared with the exact ones and with those obtained within RPA and Self Consistent RPA. We find that it is possible to improve considerably upon RPA by still keeping the formalism simple enough to be applied in more realistic cases without a too serious computational effort.
We analyze an exactly solvable many-level pairing model (the picket-fence model) in a boson formalism. By making use of a Marumori-type mapping procedure, we construct a boson image of the model Hamiltonian. Only two- and four-boson operators are included in the boson expansion. We show that the physical spectrum of this Hamiltonian coincides with the exact one, and, in particular, we recover Richardson's equations. We apply this boson formalism to search for possible extensions of the random-phase approximation (RPA). We construct renormalized RPA-type equations in correspondence with different expansions of the boson operators. We discuss two extensions of the RPA and test them by comparing their predictions with the exact results and with those obtained within the RPA and the self-consistent RPA.
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Working within an exactly solvable three level model a boson Hamiltonian is defined via a mapping procedure and its expansion truncated at four-boson terms. The resulting spectrum is found in good agreement with the exact one. We discuss an extension of the random-phase approximation (RPA) based on this boson formalism. Nonlinear RPA-type equations are constructed and solved iteratively. The new solutions gain in stability with respect to the RPA ones. We perform diagonalizations of the boson Hamiltonian in restricted spaces; approximate spectra exhibit an improved quality with increasing the size of these spaces. Special attention is addressed to the problem of the anharmonicity of the spectrum.
The random-phase approximation (RPA) is examined in a boson formalism and special attention is focused on the problem of the violation of the Pauli principle affecting this theory. A mapping technique is discussed and the boson image of a two-body fermion Hamiltonian is constructed. Within the boson space so defined, an extension of RPA is proposed where structure and energy of the ground state, as well as those of the one-phonon states, are analyzed with regard to the role of the spurious components associated with a violation of the Pauli principle. A multistep minimization procedure for the determination of a better ground state is also examined. Numerical tests are performed within the Lipkin-Meshkov-Glick model.