Multipole expansion for magnetic structures: A generation scheme for a symmetry-adapted orthonormal basis set in the crystallographic point group

Identifying the order parameter for a magnetic phase is useful for a deeper understanding of physical phenomena, such as the anomalous Hall effect, electromagnetic effects, and optical responses. Here, the authors provide an intuitive and general procedure to deduce the proper order parameters of magnetic phases. The procedure is based on multipole expansion and the introduction of a novel concept, the virtual cluster. The method facilitates the understanding of order parameters in antiferromagnets and provides themissing link between order parameters and macroscopic phenomena.