The many levels pairing Hamiltonian for two pairs

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. The associated solutions of the Richardson's equations are classified in terms of a number $v_l$, which reduces to the seniority $v$ in the limit of large values of the pairing strength $G$ and yields the number of pairs not developing a collective behaviour, their energy remaining finite in the $G\to \infty$ limit. We express analytically, through the moments of the single particle levels distribution, the collective mode energy and the two critical values $G_{\mathrm{cr}}^{+}$ and $G_{\mathrm{cr}}^{-}$ of the coupling which can exist on a single particle level with no pair degeneracy. Notably $G_{\mathrm{cr}}^{+}$ and $G_{\mathrm{cr}}^{-}$ merge when the number of single particle levels goes to infinity, where they coincide with the $G_{\mathrm{cr}}$ (when it exists) of a one pair system, not envisioned by the Richardson theory. In correspondence of $G_{\mathrm{cr}}$ 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 finding out that the former, for $G>G_{\mathrm{cr}}^{-}$, come close to the BCS predictions, whereas the latter display a divergence at $G_{\mathrm{cr}}$, signaling the onset of a long range off-diagonal order in the system.

35 pages, 6 figures, 2 tables, to be published in EPJA

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Cited by in corpus (3)

• Gravity assisted solution of the mass gap problem for pure Yang-Mills fields
• The multilevel pairing Hamiltonian versus the degenerate case
• Scaling properties of the pairing problem in the strong coupling limit