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Physics Letters B

Extended Bethe-Richardson-Gaudin ansatzes for the solution of a mean-field plus separable pairing model with three non-degenerate j-orbits

Feng Pan, Yingxin Wu, Shengze Guan, Zhibo Qu, L. R. Dai, Jerry Paul Draayer

Published Aug 2, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.1016/j.physletb.2022.137362

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A general procedure, based on the Bethe ansatz, is proposed for finding algebraic solutions for low-lying J=0J=0J=0 states of 2k2k2k nucleons interacting with one another through a T=1T=1T=1 charge-independent pairing interaction. The results provided by Richardson are shown to be valid for up to two pairs, k< 2;k<~2;k< 2; expressions are given here for up to three pairs, k< 3.k<~3.k< 3. The results show that a set of highly nonlinear equations must be solved for k> 3.k>~3.k> 3.
A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For kkk pairs of valence nucleons on nnn different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one $(k+1)\ifmmode\times\else\texttimes\fi{}(k+1)$ and one $(n\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(k+1)$ matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem, which by extension, can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multiparticle systems.
A new step-by-step diagonalization procedure for evaluating exact solutions of the nuclear deformed mean-field plus pairing interaction model is proposed via a simple Bethe ansatz in each step from which the eigenvalues and corresponding eigenstates can be obtained progressively. This new approach draws upon an observation that the original one- plus two-body problem in a kkk-particle Hilbert subspace can be mapped onto a one-body grand hard-core boson picture that can be solved step by step with a simple Bethe ansatz known from earlier work. Based on this new procedure, it is further shown that the extended pairing model for deformed nuclei [Feng Pan, V. G. Gueorguiev, and J. P. Draayer, Phys. Rev. Lett. 92, 112503 (2004)] is similar to the standard pairing model with the first step approximation, in which only the lowest energy eigenstate of the standard pure pairing interaction part is taken into consideration. Our analysis shows that the standard pairing model with the first step approximation displays similar pair structures of the first few exact low-lying states of the model, which, therefore, provides a link between the two models.