Pairing correlations: II. Exact model ground-state results for generalised ladders

In this series of papers, the so-called ground-state version of the [exp(S) or] coupled-cluster formalism (CCF) of quantum many-body theory is applied to the general problem of pairing correlations within a many-body system of identical fermions. In this second work in the series we restrict ourselves to exact calculations and concentrate on analytic solutions to the generalised ladder approximations formulated in the first paper. We focus attention on the particular model case of a general (non-local) separable potential, and work within the so-called complete ladder (CLAD) approximation which was shown in the earlier paper to be the CCF formulation of the well-known Galitskii approximation. We show how the CLAD approximation reduces in this case to a highly non-trivial pair of coupled nonlinear integral equations for the four-point correlation function, S2, which provides a measure of the two-particle/two-hole component in the true "ground-state" wave-function. In the further derivation of exact analytic solutions for both S2 and the corresponding "ground-state" energy, we also see how various types of composite pairs within the many-body medium manifest themselves as "virtual (de-)excitations". We thus show how our CCF provides an efficient and unified framework in which to describe all aspects of pairing, such as: (i) a possible free bound pair and its gradual approach to "dissolution" as the density is increased; (ii) the possible appearance of a second bound pair of predominantly hole-like quasi-particles above some lower critical density (which depends on the total momentum of the pair); (iii) the unstable but bound resonant pairs that can exist for densities above a comparable upper critical density at which the two previous types of real bound pairs have "dissolved"; and (iv) Cooper pairs. Even though each of these composite pairs leads to a new "condensed-pair phase" of lower energy, we further show that our so-called ground-state CCF leads only to the fluid-like state of uncondensed particles. In a third paper in this series we use the solutions obtained here as input to the analogous excited-state version of the CCF, and show how these various composite pairs materialise as "negative energy (de-)excitations".