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Physical Review C

Boson expansion techniques, the Pauli principle, and the quasiparticle random phase approximation phase transition

Jorge G. Hirsch, Peter Oto Hess, Osvaldo Civitarese

Published Oct 26, 1999• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.60.064303

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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.
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.
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.
Proton-neutron random phase approximation studied by the Lipkin-Meshkov-Glick model in the SU(2) ×\times× SU(2)
We study the proton-neutron RPA with an extended Lipikin-Meshkov-Glick model. We pay attention to the effect of correlated ground state and the case in which neutron and proton numbers are different. The effect of the correlated ground state are tested on the basis of quasi-boson approximation. We obtain the result that RPA excitation energies and transition strengths are in a good agreement with the exact solution up to a certain strength of the particle-particle interaction. However, the transition strength becomes worse if we consider the case in which neutron and proton numbers are different even at a weak particle-particle interaction.
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.
An extended Lipkin-Meshkov-Glick model for testing the proton-neutron random-phase approximation (pnRPA)(pn\mathrm{RPA})(pnRPA) method is developed, taking into account explicitly proton and neutron degrees of freedom. Besides the proton and neutron single-particle terms two types of residual proton-neutron interactions, one simulating a particle-particle and the other a particle-hole interaction, are included in the model Hamiltonian so that the model is exactly solvable in an isospin $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ basis. The behavior of the first excited (collective) state obtained by (i) exact diagonalization of the Hamiltonian matrix and (ii) with the pnRPA\mathrm{pn}\mathrm{RPA}pnRPA is studied as a function of the model parameters and the two results are compared with each other. Furthermore, charge-changing operators simulating nuclear beta decay and their action on eigenfunctions of the model Hamiltonian are defined and transition amplitudes of them are calculated using exact, the Tamm-Dancoff, and pnRPA\mathrm{pn}\mathrm{RPA}pnRPA eigenfunctions.