On completeness of Bethe Ansatz solutions for sl(2) Richardson–Gaudin systems

T. Skrypnyk

Jon Links

Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson–Gaudin system it will be argued that, for generic values of the system’s coupling parameters, the Bethe states are complete. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothesis, and is generalisable to a wider class of systems.

Feng Pan, Yingxin Wu, Shengze Guan +3 more

Based on the Bethe-Richardson-Gaudin ansatz for the standard pairing model, extended Bethe- Richardson-Gaudin ansatzes for eigenvectors of a spherical mean-field plus separable pairing model with three non-degenerate j-orbits are proffered. It is shown that the number of variables appearing in the general extended ansatz eigenvectors for given number of pairs N is N(N+1)/2. More importantly, when one of the j orbits is 1/2, there are only 2N variables involved in the alternative extended ansatz eigenvectors, which, like the standard pairing model, can be solved efficiently. Numerical results for an application of the model in the ds-shell up to its half-filling are presented, which serves to validate the procedure and illustrates the completeness of the solutions it renders.

Ferenc Woynarovich

V. Tarasov

Yibing Shen, Jon Links

An integrable model describing the interconversion between Bose-Einstein condensates of atomic and molecular degrees of freedom is studied. An exact solution for this model has been previously derived via the algebraic Bethe Ansatz formalism. Here we derive an alternative solution of the Richardson-Gaudin form, and highlight some of the implications of the result.