Boson mappings and four-particle correlations in algebraic neutron-proton pairing models

Neutron-proton pairing correlations are studied within the context of two solvable models, one based on the algebra SO(5) and the other on the algebra SO(8). In both of these models, particles interact in $L=0\mathrm{}$ pairs only. Boson-mapping techniques are applied to these models and shown to provide a convenient methodological tool both for solving such problems and for gaining useful insight into general features of pairing. We first focus on the SO(5) model, which involves generalized $T=1\mathrm{}$ pairing. Neither boson mean-field methods nor fermion-pair approximations are able to describe in detail neutron-proton pairing in this model. The analysis suggests, however, that the boson Hamiltonian obtained from a mapping of the fermion Hamiltonian contains a pairing force between bosons, pointing to the importance of boson-boson (or equivalently four-fermion) correlations with isospin $T=0\mathrm{}$ and spin $S=0\mathrm{}$. These correlations are investigated by carrying out a second boson mapping. Closed forms for the fermion wave functions are given in terms of the fermion-pair operators. Similar techniques are applied—albeit in less detail—to the SO(8) model, involving a competition between $T=1\mathrm{}$ and $T=0\mathrm{}$ pairing. Conclusions similar to those of the SO(5) analysis are reached regarding the importance of four-particle correlations in systems involving neutron-proton pairing.