Test of the proton-neutron random-phase approximation method within an extended Lipkin-type model

An extended Lipkin-Meshkov-Glick 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 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 their transition amplitudes are calculated using exact, Tamm-Dancoff and pnRPA eigenfunctions. (authors)