An extended Lipkin-type model with residual proton-neutron interaction

Abstract

The Lipkin-Meshkov-Glick (LMG) model is extended to explicitly take into account the proton and neutron degrees of freedom. The proton and neutron Hamiltonians are taken to be of the LMG form, and, in addition, a residual proton-neutron interaction is include. Exact solutions in an SU(2) ⊗ SU(2) basis as well as the RPA solutions for the energy spectrum of the model Hamiltonian are obtained. The spectrum of the exact solutions is degenerate in the limit of no proton-neutron residual interaction, but this degeneracy is totally removed when this type of residual interaction is turned on. The spectrum obtained with RPA is compressed or expanded, as compared to the LMG model with the same total number of nucleons ( N), depending of the relative magnitudes of the like- and unlike-particle residual interactions. Furthermore, the behavior of the first collective excited state is studied as function of the interaction parameters of the model using the exact and RPA methods. At a given N, it was found that the RPA result is closer to the exact one when the like- and unlike-particle residual interactions are both taken into account in the model Hamiltonian, as compared to the case when residual interaction of only one type (e.g. either like- or unlike-particle type) is involved.