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

Abstract

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.

Program summary

Program Title: PointGroupNRG

CPC Library link to program files: https://doi.org/10.17632/hjwmt6cc55.1

Developer's repository link: https://github.com/aitorcf/PointGroupNRG

Licensing provisions: GPLv3

Programming language: Julia

Journal reference of previous version: Comput. Phys. Commun. 296 (2024), 109032, https://doi.org/10.1016/j.cpc.2023.109032

Does the new version supersede the previous version?: Yes.

Reasons for the new version: Extension.

Nature of problem: Numerical renormalization group (NRG) calculations for realistic models are computationally expensive, mainly due to their hard scaling with the number of orbital and spin configurations available for the electrons. Symmetry considerations reduce the computational cost of the calculations by exploiting the block structure of the operator matrix elements and by removing the redundancy in the symmetry-related matrix elements. Existing codes implement continuous symmetries, which are not generally and/or straightforwardly applicable to systems where spin-orbit and crystal-field effects need to be taken into account.

Solution method: The first version of the code [1] introduced finite point group symmetries together with particle conservation and spin isotropy, useful for systems with strong crystal-field effects but negligible spin-orbit coupling. This new version also includes total angular momentum conservation and double group symmetries, together with particle conservation. This allows to deal with magnetic impurity systems with strong spin-orbit coupling. To make the code more versatile in handling systems of this type, we have added the possibility to use ionic models.