  We present a systematic method to automatically generate symmetry-adapted
magnetic structures for given crystal structure and general propagation vector
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 multipole expansion, which was demonstrated only for the propagation
vector k=0 [M.-T. Suzuki et al., Phys. Rev. B 99, 174407 (2019)]. The
symmetry-adapted magnetic structures characterized with an ordering vector 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 k.
This method provides all magnetic bases compatible with irreducible
representations under k-group for given crystal structure and wave vector k.
Multiple-k magnetic structures are derived from superposition of single-k
magnetic bases related with space group symmetry. We apply the scheme to deduce
the magnetic structures of $\alpha$-Mn and Co$M_{3}$S$_6$ (M = Nb, Ta), in
which 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.

Strongly Correlated Electrons