Generation of modulated magnetic structures based on cluster multipole expansion: Application to α -Mn and CoM3S6

We present a systematic method to automatically generate symmetry-adapted magnetic structures for a given crystal structure and general propagation vector $\mathbit{k}$ as an efficient approach of the analysis of complex modulated magnetic structures. The method is developed as an extension of the generation scheme based on the multipole expansion, which was demonstrated only for the propagation vector $\mathbit{k}=\mathbit{0}$ [M.-T. Suzuki et al., Phys. Rev. B 99, 174407 (2019)]. The symmetry-adapted magnetic structures characterized with an ordering vector $\mathbit{k}$ are obtained by mapping the multipole magnetic alignments on a virtual cluster to the periodic crystal structure with the phase factor for the wave vector $\mathbit{k}$. This method provides all magnetic bases compatible with irreducible representations under a $\mathbit{k}$ group for a given crystal structure and wave vector $\mathbit{k}$. Multiple-$\mathbit{k}$ magnetic structures are derived from a superposition of single-$\mathbit{k}$ magnetic bases related to the space group symmetry. We apply the scheme to deduce the magnetic structures of $\ensuremath{\alpha}$-Mn and $\mathrm{Co}{M}_{3}{\mathrm{S}}_{6}$ ($M=\mathrm{Nb}$, Ta), in which the large anomalous Hall effect has recently been observed in antiferromagnetic phases, and identify the magnetic structures inducing anomalous Hall effect without net magnetization. The physical phenomena originating from emergent multipoles in the ordered phases are also discussed based on the Landau theory.