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Physical Review C

Random phase approximation studies in the fermion dynamical symmetry model: SO(8)

G. Wenes, Joseph N. Ginocchio

Published Jun 1, 1989• 1 citation
Publisher

Abstract

The SO(8) dynamical symmetry limit of the Fermion dynamical symmetry model is used to study various aspects of the quasiparticle random phase approximation. The zero frequency modes of the general quasiparticle random phase approximation are given in analytical form. The quasiparticle random phase approximation excitation energy and transition probability are derived in analytical form in the spherical limit of the general SO(8) Hamiltonian.

DOI
10.1103/physrevc.39.2426

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The SO(8) dynamical symmetry limit of the Fermion dynamical symmetry model is used to study various aspects of the quasiparticle random phase approximation. The zero frequency modes of the general quasiparticle random phase approximation are given in analytical form. The quasiparticle random phase approximation excitation energy and transition probability are derived in analytical form in the spherical limit of the general SO(8) Hamiltonian.
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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.
In the present paper a number conserving random phase approximation is derived as a special case of the recently developed random phase approximation in general symmetry projected quasiparticle mean fields. All the occurring integrals induced by the number projection are performed analytically after writing the various overlap and energy matrices in the random phase approximation equation as polynomials in the gauge angle. In the limit of a large number of particles the well-known pairing vibration matrix elements are recovered. We also present a new analytically number projected variational equation for the number conserving pairing problem.
The one-dimensional Fermi gas with attractive \ensuremath{\delta} interaction is treated in the quasiparticle random-phase approximation at zero temperature. The collective modes are evaluated numerically in the high-density (weak-coupling) and in the low-density (strong-coupling) case. Whereas in the weak-coupling limit the ordinary (particle-hole) random-phase approximation is approached for low momenta, the collective mode in the strong-coupling limit reproduces the Bogoliubov mode for the weakly interacting gas of particle-particle pairs (bosons). We find a smooth evolution of the collective modes from weak to strong coupling. Analytical approximations for the dispersion relation in the long-wavelength limit are derived for both weak and strong coupling. \textcopyright{} 1996 The American Physical Society.
In this Brief Report we investigate excited yrast states under random interactions in the framework of the fermion dynamical symmetry model, for the ensemble with spin-0 ground states. Interesting correlations are seen between R6{R}_{6}R6​ and R4{R}_{4}R4​ (where ${R}_{I}\ensuremath{\equiv}{E}_{{I}_{1}^{+}}/{E}_{{2}_{1}^{+}}$) by using the Mallmann plot, for cases with both SP(6) symmetry and SO(8) symmetry.
A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.