The Heine-Stieltjes correspondence and a new angular momentum projection for many-particle systems

Abstract

A new angular momentum projection for systems of particles with arbitrary spins is formulated based on the Heine-Stieltjes correspondence, which can be regarded as the solutions of the mean-field plus pairing model in the strong pairing interaction G ->Infinity limit. Properties of the Stieltjes zeros of the extended Heine-Stieltjes polynomials, of which the roots determine the projected states, and the related Van Vleck zeros are discussed. The electrostatic interpretation of these zeros is presented. As examples, applications to n nonidentical particles of spin-1/2 and to identical bosons or fermions are made to elucidate the procedure and properties of the Stieltjes zeros and the related Van Vleck zeros. It is shown that the new angular momentum projection for n identical bosons or fermions can be simplified with the branching multiplicity formula of U(N) supset O(3) and the special choices of the parameters used in the projection. Especially, it is shown that the solutions for identical bosons can always be expressed in terms of zeros of Jacobi polynomials. However, unlike non-identical particle systems, the n-coupled states of identical particles are non-orthogonal with respect to the multiplicity label after the projection.

14 pages LaTeX with no figure

References in corpus (5)

• Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence
• Richardson-Gaudin integrability in the contraction limit of the quasispin
• The Lipkin-Meshkov-Glick model as a particular limit of the SU(1,1) Richardson-Gaudin integrable models
• Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models
• New exact solutions of the standard pairing model for well-deformed nuclei

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• Exact solution of the two-axis countertwisting Hamiltonian
• Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin
• Exact solution of the two-axis two-spin Hamiltonian