Hyperspin Formulation of a Modified Pairing Treatment of the Bose Fluid

Abstract

A modified pairing treatment is developed for handling the $N$-boson problem with repulsive particle-particle interaction. The nonpairing portion of the many-boson Hamiltonian is eliminated, to first order, by a canonical transformation. The resultant Hamiltonian contains an additional attractive particle-particle interaction, whenever there is a Bose condensation in the k=0 single-particle state. The Bose condensate plays the same role in mediating this interaction that the phonon field does in mediating the attractive interaction between electrons in a superconductor. The strength of the condensate-induced attractive interaction is approximately double that of the direct repulsive interaction. The region of $k$ space over which the attractive interaction operates depends on $N_0$, the number of bosons in the condensate, this region vanishing when $N_0$ does. For convenience, the problem is formulated in terms of hyperspin, in analogy with the isospin treatment of superconductivity. The ground state is characterized by two order parameters ${\Delta }_1$ and ${\Delta }_2$, and an energy $\Gamma$ (proportional to $N_0$), which can be obtained by solving a set of coupled integral equations. It is shown that there is no gap, at low energies, in the single-particle excitation spectrum. Aside from these single-particle excited states, there are no other collective oscillations. The hyperspin formulation is extended to finite temperatures. There are, however, certain difficulties with this extension.