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arXiv:0710.3941 · Physical Review C

Solutions of nuclear pairing

A. B. Balantekin, Yamaç Pehlivan

Published Nov 14, 2007• 18 citations
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Abstract

We give the exact solution of orbit dependent nuclear pairing problem between two nondegenerate energy levels using the Bethe ansatz technique. Our solution reduces to previously solved cases in the appropriate limits including Richardson's treatment of reduced pairing in terms of rational Gaudin algebra operators.
DOI
10.1103/physrevc.76.051001

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Quantum invariants of the orbit dependent pairing problem are identified in the limit where the orbits become degenerate. These quantum invariants are simultaneously diagonalized with the help of the Bethe ansatz method and a symmetry in their spectra relating the eigenvalues corresponding to different number of pairs is discussed. These quantum invariants are analogous to the well known rational Gaudin magnet Hamiltonians which play the same role in the reduced pairing case (i.e., orbit independent pairing with non degenerate energy levels). It is pointed out that although the reduced pairing and the degenerate cases are opposite of each other, the Bethe ansatz diagonalization of the invariant operators in both cases are based on the same algebraic structure described by the rational Gaudin algebra.
Exact solution of the nuclear mean-field plus separable pairing model is reexamined. New auxiliary constraints for solving the Bethe ansatz equations of the model are proposed. By using these auxiliary constraints, the Bethe ansatz form of eigenvectors of the mean-field plus separable pairing Hamiltonian with nondegenerate single-particle energies and nondegenerate separable pairing strengths purposed previously is verified. Since the solutions of the model with one- and two-orbit cases are known, verification of the solutions for these two special cases is made. To demonstrate structure and features of the solution, the model with three orbits in the dsdsds shell is taken as a nontrivial example, of which two-pair results and the ground state of the three-pair case are provided explicitly. Since the number of equations involved increases with the number of orbits and pairs, to solve these equations for a large number of orbits and pairs seems still difficult.
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.