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Physical Review C

Particle number fluctuations in the quasiparticle random-phase approximation and renormalized quasiparticle random-phase approximation

A. Mariano, Jorge G. Hirsch

Published Jun 1, 1998• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.57.3015

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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 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.
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.
Exact results of pair transfer probabilities for the Richardson model with equidistant or random level spacing are presented. The results are then compared either to particle-particle random-phase approximation (ppRPA) in the normal phase or quasiparticle random-phase approximation (QRPA) in the superfluid phase. We show that both ppRPA and QRPA are globally well reproducing the exact case although some differences are seen in the superfluid case. In particular, the QRPA overestimates the pair transfer probabilities to excited states in the vicinity of the normal-superfluid phase transition, which might explain the difficulty in detecting collective pairing phenomena as, for example, the giant pairing vibration. The shortcoming of QRPA can be traced back to the breaking of particle number that is used to incorporate pairing. A method based on direct diagonalization of the Hamiltonian in the space of two quasiparticles projected onto good particle number is shown to improve the description of pair transfer probabilities in superfluid systems.
The onset of instabilities, in the standard proton-neutron quasiparticle random phase approximation (QRPA), is investigated by using different boson mappings and trial states belonging to a representation of coherent states. The model describes pairing and proton-neutron interactions. It is shown that an exact mapping of the Hamiltonian is needed to describe correctly the QRPA phase transition, i.e., that the algebra of all operators participant in the Hamiltonian should be preserved by the boson mapping. Spurious components of the wave functions, which appear if the Pauli principle is violated, are isolated by construction.