Effective spin model in momentum space: Toward a systematic understanding of multiple-$Q$ instability by momentum-resolved anisotropic exchange interactions

Abstract

Multiple-Q magnetic states, such as a skyrmion crystal, become a source of unusual transport phenomena and dynamics. Recent theoretical and experimental studies clarify that such multiple-Q states ubiquitously appear under different crystal structures in metals and insulators. Toward a systematic understanding of the formation of the multiple-Q states in various crystal systems, in this theoretical study, we present a low-energy effective spin model with anisotropic exchange interactions in momentum space. We summarize specific six symmetry rules for nonzero symmetric and antisymmetric anisotropic exchange interactions in momentum space, which are regarded as an extension of Moriya's rule. According to the rules, we construct the effective spin model for tetragonal, hexagonal, and trigonal magnets with crystal-and momentum-dependent anisotropic exchange interactions based on magnetic representation analysis. Furthermore, we describe the origin of the effective anisotropic exchange interactions in itinerant magnets by perturbatively analyzing a multi-band periodic Anderson model with the spin-orbit coupling. We apply the effective spin model to an itinerant magnet in a P 6/mmm crystal and find various multiple-Q states with a spin scalar chirality in the ground state. Our results provide a foundation of constructing effective phenomenological spin models for any crystal systems hosting the multiple-Q states, which will stimulate further exploration of exotic multiple-Q states in materials with the spin-orbit coupling.

Skyrmion crystal under $D_{3h}$ point group: Role of out-of-plane Dzyaloshinskii-Moriya interaction

“…Although the interactions between the further-neighboring spins play an important role in the RKKY system, only the out-of-plane DM interaction is present due to horizontal mirror symmetry under D 3h point group. The instabilities toward the SkXs based on a similar effective spin model with a particular set of the momentum-resolved interactions under other point groups have been investigated in both insulators [81,111,118,144,145] and metals [92,94,[104][105][106]110]. Hereafter, we set J = 1 as the energy unit and change D as a phenomenological parameter.…”

Skyrmion crystals in centrosymmetric triangular magnets under hexagonal and trigonal single-ion anisotropy

“…Although the long-range nature of the interaction like the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [81,82,83] in itinerant magnets is different from the short-range one in frustrated insulating magnets, an effective multi-spin interaction arising from the Fermi surface instability gives rise to the groundstate SkX even without the DM interaction [78]. Similar to the frustrated magnets, it was shown that the effect of single-ion anisotropy [84,85,86,87], two-ion anisotropy [88,89,90,91,92,93], circularly polarized microwave field [94], and thermal fluctuations [95,96] enhance the stability of the SkX. It was also shown that the introduction of the DM or antisymmetric spin-orbit interaction in itinerant magnets leads to the SkX formation [43,97,98,99,100].…”

Helicity locking of square skyrmion crystal in a centrosymmetric lattice system without vertical mirror symmetry

“…Meanwhile, magnetic anisotropy that arises from the rotational and mirror symmetry breakings in the specific lattice structures while keeping the inversion symmetry also becomes a source of the SkXs [79]. Although the form of the spin interactions depends on a way of breakings of the lattice symmetry, such an effect can appear in any discrete lattice systems via the spin-orbit coupling.…”