Test of the pnRPA method within an extended Lipkin-type model

An extended Lipkin-Meshkov-Glick (LMG) model for testing the proton-neutron Random Phase Approximation (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 SU(2) x 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, is studied as function of the model parameters and the two results 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, and the Tamm-Dancoff and pnRPA eigenfunctions. (authors)