Effective and efficient algorithm for the Wigner rotation matrix at high angular momenta

Abstract

The Wigner rotation matrix ($d$ function), which appears as a part of the angular-momentum-projection operator, plays a crucial role in modern nuclear-structure models. However, it is a longstanding problem that its numerical evaluation suffers from serious errors and instability, which hinders precise calculations for nuclear high-spin states. Recently, Tajima [Phys. Rev. C 91, 014320 (2015)] has made a significant step toward solving the problem by suggesting the high-precision Fourier method, which however relies on formula-manipulation softwares. In this paper we propose an effective and efficient algorithm for the Wigner $d$ function based on the Jacobi polynomials. We compare our method with the conventional Wigner method and the Tajima Fourier method through some testing calculations, and demonstrate that our algorithm can always give stable results with similar high-precision as the Fourier method, and in some cases (for special sets of $j,m,k$, and $\ensuremath{\theta}$) ours are even more accurate. Moreover, our method is self-contained and less memory consuming. By taking the $^{156}\mathrm{Dy}$ yrast band as an example, we show that with the $d$ function calculated by our proposed Jacobi method in the angular-momentum projector, the realistic projected-shell-model calculation can be aggressively extended to the high-spin region where the conventional Wigner method collapses completely. A related testing code and subroutines for the three algorithms of $d$ function are provided as Supplemental Material in the present paper.