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AIP conference proceedings

Bethe ansatz technique in shell model

A. B. Balantekin, Yamaç Pehlivan, Sun‐Chan Jeong, Y. Utsuno, T. Motobayashi, A. Bracco

Published Jan 1, 2009• 0 citations
Publisher

Abstract

We review the application of Bethe ansatz method to the shell model Hamiltonian describing s‐pairs of nucleons which interact with an orbit dependent separable pairing force. We concentrate on the special case in which the separations between the single particle energy levels are small in comparison with the pairing energy.
DOI
10.1063/1.3141680

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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 problem of one pair of identical nucleons sitting in N{\cal N}N single particle levels of a potential well and interacting through the pairing force is treated introducing, in the Hamiltonian formalism, even Grassmann variables. The eigenvectors are analytically expressed solely in terms of these with coefficients fixed by the eigenvalues and the single particle energies. For these a specific model is needed: in the case of the harmonic oscillator well, for any strength of the pairing interaction, an accurate expression is derived for both the collective eigenvalue and for those trapped in between the single particle levels. Notably the latter are labelled through an index upon which they depend parabolically.
Two advances are reviewed in the application of pairing-force theory in the nuclear shell model. The first exploits a discovery that a wide range of two-nucleon interactions conserve seniority as a good quantum number. Asa consequence, the eigenstates of a hamiltonian with such an interaction belong to irreducible representations of a compact unitary-symplectic group. This makes it possible to extend the simply-solvable models with J = 0 pairing forces to a much richer set of models and still obtain states uniquely classified by their seniority and angular momentum quantum numbers. Moreover, many of the low-lying energy levels of such models can be obtained algebraically; in technical terms, the models are in some cases completely solvable and in other cases partially solvable by algebraic methods. The second advance exploits the discovery that, in a coherent state representation, states of good nucleon number can be projected analytically from BCS vacuum and excited quasiparticle states. This makes it possible to perform calculations in a number-projected BCS basis without losing much of the advantage of working in a quasiparticle scheme.
A pairing model for nucleons, introduced by Richardson in 1966, which describes proton–neutron pairing as well as proton–proton and neutron–neutron pairing, is re-examined in the context of the quantum inverse scattering method. Specifically, this shows that the model is integrable by enabling the explicit construction of the conserved operators. We determine the eigenvalues of these operators in terms of the Bethe ansatz, which in turn leads to an expression for the energy eigenvalues of the Hamiltonian.