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

We propose a systematic method to generate a complete orthonormal basis set of multipole expansion for magnetic structures in arbitrary crystal structure. The key idea is the introduction of a virtual atomic cluster of a target crystal on which we can clearly define the magnetic configurations corresponding to symmetry-adapted multipole moments. The magnetic configurations are then mapped onto the crystal so as to preserve the magnetic point group of the multipole moments, leading to the magnetic structures classified according to the irreducible representations of the crystallographic point group. We apply the present scheme to pyrochlore and hexagonal $AB{\mathrm{O}}_3$ crystal structures and demonstrate that the multipole expansion is useful to investigate the macroscopic responses of antiferromagnets.

Figure 1

Outline of the generation procedure for a $3N_{\mathrm{atom}}$ complete basis set of multipole magnetic structure classified by the crystallographic point group.

Figure 2

Examples of virtual atomic clusters for (a) $O_h$, (b) $C_{4h}$, (c) $D_{6h}$, and (d) $C_{6v}$ point groups.

Figure 3

Magnetic configuration of the MT quadrupole in $B_g$ IREP of the $C_{4h}$ point group in the virtual atomic cluster, placing an atom labeled 1 at (0,0.3,0.3), which is obtained by Eq. (11) with ${\mathcal{Y}}_{25}$ in Eq. (18).

Figure 4

MT quadrupoles in crystals belonging to the $P4/m$ and $P{4}_2/m$ space groups, which are obtained by mapping from the $C_{4h}$ virtual atomic cluster as shown in Fig. 3.

Figure 5

Multipole magnetic structures in pyrochlore structure characterized by the IREP of $O_h$ point group and its rank. Notations of multipoles are adapted from tables in Ref. [14].

Figure 6

The multipole expansion of all-in all-out, two-in two-out, and one-in three-out magnetic structures in the pyrochlore crystal structure.

Figure 7

Multipole magnetic structures in $AB{\mathrm{O}}_3$ characterized by the IREP of the $C_{6v}$ point group and its rank. The notations of multipoles are adapted from tables in Ref. [14].