Numerical renormalization group calculations for magnetic impurity systems with spin-orbit coupling and crystal-field effects

Exploiting symmetries in the numerical renormalization group (NRG) method significantly enhances performance by improving the accuracy, increasing the computational speed, and optimizing the memory efficiency. Published codes focus on continuous rotations and unitary groups, which generally are not applicable to systems with strong crystal-field effects. The PointGroupNRG code implements symmetries related to discrete rotation groups, which are defined by the user in terms of Clebsch-Gordan coefficients, together with particle conservation and spin rotation symmetries. In this paper we present a new version of the code that extends the available finite groups, previously limited to simply reducible point groups, in a way that all point and double groups become accessible. It also includes the full spin-orbital rotation group. Moreover, to improve the code's flexibility for impurities with complex interactions, this new version allows to choose between a standard Anderson Hamiltonian for the impurity or, as another novel feature, an ionic model that requires only the spectrum and the impurity Lehmann amplitudes.在数值重整化群（numerical renormalization group, NRG）方法中利用对称性策略，可通过提升计算精度、加快运算速率以及优化内存使用效率，显著改善算法性能。已公开的相关代码多聚焦于连续旋转群与幺正群，这类方法通常无法适用于存在强晶体场效应的物理体系。PointGroupNRG代码实现了与离散旋转群相关的对称性支持，用户可通过克莱布希-高登系数（Clebsch-Gordan coefficients）定义离散旋转群，同时兼容粒子守恒与自旋旋转对称性。本文提出了该代码的新版本，将原本仅局限于简单可约点群的有限群支持范围进行扩展，如今可覆盖所有点群与双点群。该新版本还完整支持自旋轨道旋转群。此外，为提升代码针对具有复杂相互作用的杂质体系的适配灵活性，新版本提供了两种可选模型：其一为针对杂质的标准安德森（Anderson）哈密顿量，其二为一项全新特性——仅需能谱与杂质莱曼（Lehmann）振幅的离子模型。