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arXiv:nucl-th/0405002 · Physical Review C

Self-consistent random phase approximation and the restoration of symmetries within the three-level Lipkin model

Doru Sabin Delion, Peter Schuck, J. Dukelsky

Published Dec 13, 2005• 25 citations
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Abstract

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.

DOI
10.1103/physrevc.72.064305

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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 study an extended Lipkin-Meshkov-Glick model that permits a transition to a deformed phase with a broken continuous symmetry. Unlike simpler models, one sees a persistent zero-frequency Goldstone mode past the transition point into the deformed phase. We found that the RPA formula for the correlation energy provides a useful correction to the Hartree-Fock energy when the number of particle N satisfies N>3,N>3,N>3, and becomes accurate for large N. We conclude that the RPA correlation energy formula offers a promising way to improve the Hartree-Fock energy in a systematic theory of nuclear binding energies.
The validity of the random-phase approximation (RPA) in broken-symmetry bases is tested in an appropriate many-body system for which exact solutions are available. Initially the regions of stability of the self-consistent quasiparticle bases in this system are established and depicted in a 'phase' diagram. It is found that only stable bases can be used in an RPA calculation. This is particularly true for those RPA modes which are not associated with the onset of instability of the basis; the authors see that these modes do not describe any excited state when the basis is unstable, although from a formal point of view they remain acceptable. The RPA does well in a stable broken-symmetry basis provided one is not too close to a point where a phase transition occurs. This is true for both energies and matrix elements.
The random phase approximation (RPA), the renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) methods are applied to calculate the wave functions of the ground and excited states of an exactly solvable model. The approximated wave functions are expanded in the basis of the exact solutions. It is found that, when the RPA collapses, the RPA wave functions are orthogonal to the exact solutions while the RRPA and SCRPA ones have small but finite overlaps with the exact results. In spite of the apparently good agreement between the results of the RRPA, the SCRPA, and the exact solution, for the energy of the first excited state beyond the point of collapse, it is found that these approximations do not correctly describe the exact wave functions.
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 consistency condition is tested within the particle-particle random-phase approximation (RPA), renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) making use of the Richardson model of pairing. The two-particle separation energy is calculated in two ways, namely, as the energy of the first addition mode, which adds two particles to a core with NNN particles, and as the energy of the first removal mode, which removes two particles from the N+2N+2N+2 particle system to get back to the same NNN-particle core. The corresponding transitions generated by the pairing operators are also calculated. It is found that the results obtained in these two ways of calculations are close to each other only at large values of particle number NNN and/or small interaction strength. At $N\ensuremath{\le}10$ for a given value of the interaction strength, the discrepancy between the results obtained in two ways of calculations within the SCRPA is much smaller than those given by the RPA and RRPA.