Eur. Phys. J. A (2004) 22: 377-390
https://doi.org/10.1140/epja/i2004-10043-7

The many levels pairing Hamiltonian for two pairs

¹ Dipartimento di Fisica Teorica, Università di Torino, Torino, Italy
² Dipartimento di Fisica, Università di Genova, Genova, Italy

^* e-mail: barbaro@to.infn.it

Received: 26 February 2004
Accepted: 1 July 2004
Published online: 11 November 2004

Abstract

We address the problem of two pairs of fermions living on an arbitrary number of single-particle levels of a potential well (mean field) and interacting through a pairing force in the framework of the Richardson equations. The associated solutions are classified in terms of a number v_l, which reduces to the seniority v in the limit of a large pairing strength G and yields the number of pairs not developing a collective behaviour, their energy remaining finite in the G↦∞ limit. We express analytically, through the moments of the single-particle levels distribution, the collective mode energy and the two critical values G _cr ⁺ and G _cr ^- of the coupling which can exist on a single-particle level with no pair degeneracy. Notably G _cr ⁺ and G _cr ^-, when the number of single particle levels goes to infinity, merge into the critical coupling of a one-pair system G _cr (when it exists), which is not envisioned by the Richardson theory. In correspondence of G _cr, the system undergoes a transition from a mean-field- to a pairing-dominated regime. We finally explore the behaviour of the excitation energies, wave functions and pair transfer amplitudes versus G finding out that the former, for G > G _cr ^-, come close to the BCS predictions, whereas the latter display a divergence at G _cr, signaling the onset of a long-range off-diagonal order in the system.