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Physical Review C

Boson-mapping-based extension of the random-phase approximation in a three-level Lipkin model

M. Grasso, F. Catara, Michelangelo Sambataro

Published Dec 4, 2002• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.66.064303

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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.
Working within an exactly solvable 3 level model, we discuss am extension of the Random Phase Approximation (RPA) based on a boson formalism. A boson Hamiltonian is defined via a mapping procedure and its expansion truncated at four-boson terms. RPA-type equations are then constructed and solved iteratively. The new solutions gain in stability with respect to the RPA ones. We perform diagonalizations of the boson Hamiltonian in spaces containing up to four-phonon components. Approximate spectra exhibit an improved quality with increasing the size of these multiphonon 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.
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.
We show that it is possible to restore broken symmetries associated with the Goldstone mode within the self-consistent random phase approximation (SCRPA) applied to the three-level Lipkin model. To do this, it is necessary to include the so-called scattering terms in the RPA operator. We determine one- and two-body densities as very convergent expansions in terms of the generators of the RPA basis. We show that SCRPA excitations correspond to the heads of some rotational bands in the exact spectrum.
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.
It is shown that the random-phase approximation (RPA) method with its nonlinear higher generalization, which was previously considered as approximation except for a very limited case, reproduces the exact solutions of the Lipkin model. The nonlinear higher RPA is based on an equation nonlinear on eigenvectors and includes many particle-many hole components in the creation operator of the excited states. We demonstrate the exact character of solutions analytically for the particle number [Formula: see text] and, numerically, for [Formula: see text]. This finding indicates that the nonlinear higher RPA is equivalent to the exact Schrödinger equation, which opens up new possibilities for realistic calculations in many-body problems.