Exact solution of spherical mean-field plus special orbit-dependent non-separable pairing model with multi non-degenerate j-orbits

Feng Pan,Dan Zhou,Yingwen He,Siyu Yang,Yunfeng Zhang,J.P Draayer

doi:10.1016/j.physletb.2019.05.053

Feng Pan, Dan Zhou + Show 4 more

Open Access

https://doi.org/10.1016/j.physletb.2019.05.053

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Abstract

Exact solution of spherical mean-field plus a special orbit-dependent non-separable pairing Hamiltonian with multi non-degenerate j-orbits, which is related to two previously known hyperbolic Gaudin models, is explored. It is shown that the Hamiltonian with suitable constraints on the pairing interaction parameters turns to be exactly solvable. The extended one-variable Heine-Stieltjes polynomials associated to the Bethe-Gaudin-Richardson ansatz equations of the solution for any number of pairs k are determined. It is shown that the pair excitation energies can be calculated more easily than those of the separable pairing model studied previously. As examples of the solution, pairing excitation energies with the number of pairs up to the half-filling in the ds-shell with 3 j-orbits and in the pf-shell with 4 j-orbits are presented and compared with those of the mean-field plus the general separable, the special separable, and the standard pairing models. It is shown that the pairing excitation energies of the model are close to those of the mean-field plus special separable pairing or general separable pairing model.

Highlights

It is known that spherical or deformed mean-field plus the standard pairing interaction can be solved exactly by using the Bethe-Gaudin-Richardson method [1,2,3], of which the numerical solution can be calculated relatively by using the extended Heine-Stieltjes polynomial approach [4,5,6,7]
It can be shown that the Hamiltonian (1) with the special non-separable pairing (SNSP) (3) under the parameter constraints G+,− = G−,+ = 0 is related to the hyperbolic Gaudin models studied in [10,17,18,19,20,21]
Similar to the special separable pairing [10,11,12], there are 3 or 4 independent parameters adjustable in the pairing interaction matrix due to the special constraints for p = 3 or p ≥ 4 cases, where p is the number of non-degenerate j-orbits considered, which, is non-separable in general

Summary

Introduction

It is known that spherical or deformed mean-field plus the standard (equal strength) pairing interaction can be solved exactly by using the Bethe-Gaudin-Richardson method [1,2,3], of which the numerical solution can be calculated relatively by using the extended Heine-Stieltjes polynomial approach [4,5,6,7]. Exact solution of the general separable pairing (SP) model with multi degenerate orbits [8] or two non-degenerate j-orbits [9] were derived. It is shown that exact solution to a special separable pairing (SSP) interaction with multi non-degenerate orbits, like the standard pairing model (SPM), can be calculated relatively [10,11,12], while the general SP case has been analyzed in [13]. In this work, based on our recent work on exact solution of the mean-field plus orbit-dependent non-separable pairing model with two non-degenerate orbits [14], it is shown that the spherical mean-field plus orbit-dependent non-separable pairing interaction with multi non-degenerate j-orbits can be solved relatively when the pairing interaction parameters satisfy special constraints

The model and exact solution

Model application to the ds- and p f -shell

Summary