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Physical Review C

Comparative study on the validity of the renormalized random-phase approximation

Osvaldo Civitarese, M. Reboiro

Published Jun 1, 1998• 10 citations
Publisher

Abstract

The consistency of the renormalized random-phase approximation (RRPA), to treat correlations nearby a phase transition, is analyzed. The results of the RRPA and its quasiparticle version (RQRPA) for monopole and quadrupole particle-hole excitations and pairing vibrational modes, are compared with exact results and with results of the conventional quasiparticle random-phase approximation (QRPA). The calculations are performed in schematic models.

DOI
10.1103/physrevc.57.3062

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The consistency of the renormalized random-phase approximation (RRPA), to treat correlations nearby a phase transition, is analyzed. The results of the RRPA and its quasiparticle version (RQRPA) for monopole and quadrupole particle-hole excitations and pairing vibrational modes, are compared with exact results and with results of the conventional quasiparticle random-phase approximation (QRPA). The calculations are performed in schematic models.
The ground state of a many body Hamiltonian considered in the quasiparticle representation is redefined by accounting for the quasiparticle quadrupole pairing interaction. The residual interaction of the newly defined quasiparticles is treated by the quasiparticle random phase approximation (QRPA). Solutions of the resulting equations exhibit specific features. In particular, there is no interaction strength where the first root is vanishing. A comparison with other renormalization methods is presented. Application to a single [Formula: see text]-shell allows for the results interpretation by comparing them with those obtained by exact calculations.
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.
Particle number fluctuations in the quasiparticle random-phase approximation (QRPA) and the renormalized quasiparticle random-phase approximation are investigated. A closed expression for the ground state wave function is used, which allows an exact evaluation of the ground state phonon number and correlation function, being valid to any order in the backward amplitudes. In realistic calculations a sudden increase in the particle number fluctuations is found in both approximations when the particle-particle interaction approaches the value at which the QRPA collapses. This behavior is strongly correlated with the ground state quasiparticle content and can be understood as a signature of the phase transition previously found in simpler exactly solvable models.
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.
The dynamical effects of ground state correlations for excitation energies and transition strengths near the superfluid phase transition are studied in the soluble two level pairing model, in the context of the particle-particle self-consistent random phase approximation (SCRPA). Exact results are well reproduced across the transition region, beyond the collapse of the standard particle-particle random phase approximation. The effects of two-body correlation in the SCRPA are displayed explicitly.