Bosonization of the Pairing Hamiltonian

We address the problem of the bosonization of finite fermionic systems with two different approaches. First we work in the path integral formalism, showing how a truly bosonic effective action can be derived from a generic fermionic one with a quartic interaction. We then apply our scheme to the pairing hamiltonian in the degenerate case proving that, in this instance, several of the features characterizing the spontaneous breaking of the global gauge symmetry U(1) occurring in the infinite system persist in the finite system as well. Accordingly we interpret the excitations associated with the addition and removal of pairs of fermions as a quasi-Goldstone boson and the excitations corresponding to the breaking of a pair (seniority one states in the language of the pairing hamiltonian) as Higgs modes. Second, we face the more involved problem of a non-degenerate single particle spectrum, where one more kind of excitations arises, corresponding to the promotion of pairs to higher levels. This we do by solving directly the Richardson equations. From this analysis the existence emerges of critical values of the coupling constant, which signal the transition between two regimes, one dominated by the mean field physics, the other by the pairing interaction.