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arXiv:nucl-th/0312070 · Physical Review C

Solving the Richardson equations for fermions

S. Rombouts, Dimitri Van Neck, J. Dukelsky

Published Jun 11, 2004• 64 citations
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Abstract

Forty years ago Richardson showed that the eigenstates of the pairing Hamiltonian with constant interaction strength can be calculated by solving a set of nonlinear coupled equations. However, in the case of fermions these equations lead to singularities which made them very hard to solve. This paper explains how these singularities can be avoided through a change of variables making the fermionic pairing problem numerically solvable for arbitrary single-particle energies and degeneracies.

DOI
10.1103/physrevc.69.061303

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Forty years ago Richardson showed that the eigenstates of the pairing Hamiltonian with constant interaction strength can be calculated by solving a set of nonlinear coupled equations. However, in the case of fermions these equations lead to singularities which made them very hard to solve. This paper explains how these singularities can be avoided through a change of variables making the fermionic pairing problem numerically solvable for arbitrary single-particle energies and degeneracies.
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
The exact solvability of several nuclear models with non-degenerate single-particle energies is outlined and leads to a generalization of integrable Richardson-Gaudin models, like the su(2)su(2)su(2)-based fermion pairing, to any simple Lie algebra. As an example, the so(5)∼sp(4)so(5)\sim sp(4)so(5)∼sp(4) model of T=1 pairing is discussed and illustrated for the case of 64^{64}64Ge with non-degenerate single-particle energies.
This Letter provides the solution to a yet unsolved basic problem of Solid State Physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 year old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these NNN pairs via NNN parameters coupled through NNN non-linear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the NNN-pair energy depends on NNN as N(N−1)N(N-1)N(N−1) only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.