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arXiv:math-ph/0609022 · Journal of Physics A Mathematical and General

Solving the Richardson equations close to the critical points

Fernando Domínguez, C Esebbag, J. Dukelsky

Published Aug 29, 2006• 36 citations
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Abstract

We study the Richardson equations close to the critical values of the pairing strength g c , where the occurrence of divergences precludes numerical solutions. We derive a set of equations for determining the critical g values and the non-collapsing pair energies. Studying the behaviour of the solutions close to the critical points, we develop a procedure to solve numerically the Richardson equations for arbitrary coupling strength.

DOI
10.1088/0305-4470/39/37/002

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We study the Richardson equations close to the critical values of the pairing strength g c , where the occurrence of divergences precludes numerical solutions. We derive a set of equations for determining the critical g values and the non-collapsing pair energies. Studying the behaviour of the solutions close to the critical points, we develop a procedure to solve numerically the Richardson equations for arbitrary coupling strength.
Since the pioneering work of Richardson more than forty years ago, it has been known that the eigenequations for the constant pairing Hamiltonian can be reduced to a set of non-linear equations for a limited number of complex variables.Therefore the constant pairing model was said to be 'exactly solvable'.However, for fermionic particles the solution of the set of non-linear equations turned out to be hampered by uncontrollable divergencies.Here we show how one can handle these divergencies and obtain the full solution of the exactly solvable pairing models.
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 vl, 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 → ∞ 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 merge when the number of single particle levels goes to infinity, where they coincide with the Gcr (when it exists) of a one pair system, not envisioned by the Richardson theory. In correspondence of Gcr 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 − cr, come close to the BCS predictions, whereas the latter display a divergence at Gcr, signaling the onset of a long range off-diagonal order in the system. PACS numbers: 21.60.-n, 21.30 Fe, 24.10.Cn 1 1
A new technique is developed in this paper for solving Hamiltonians of the form $H=\ensuremath{\Sigma}\stackrel{}{k>0}{\ensuremath{\epsilon}}_{k}({{a}_{k}}^{+}{a}_{k}+{{a}_{\ensuremath{-}k}}^{+}{a}_{\ensuremath{-}k})\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{k,l>0}G(k,l){{a}_{k}}^{+}{{a}_{\ensuremath{-}k}}^{+}{a}_{\ensuremath{-}l}{a}_{l},$ where G(k,l)G(k, l)G(k,l) may have the forms: G(k,l)=GG(k, l)=GG(k,l)=G (independent of kkk and lll), G(k,l)=GkGlG(k, l)={G}_{k}{G}_{l}G(k,l)=Gk​Gl​, or G(k,l)=G(k,l)G(k, l)=G(k, l)G(k,l)=G(k,l) (nonseparable in kkk and lll). The eigenvalues obtained with this technique are compared with the exact solutions of Kerman, Lawson, and Macfarlane. The agreement with the exact solutions is fairly good.
We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a different set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root-finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model, and the generalized Dicke model.