Noncollinear and noncoplanar magnetic textures including Skyrmions and vortices act as emergent electromagnetic fields and give rise to novel electronic and transport properties. We here report a unified understanding of noncoplanar magnetic orderings emergent from the spin-charge coupling in itinerant magnets. The mechanism has its roots in effective multiple spin interactions beyond the conventional Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism, which are ubiquitously generated in itinerant electron systems with local magnetic moments. By carefully examining the higher-order perturbations in terms of the spin-charge coupling, we construct a minimal effective spin model composed of the bilinear and biquadratic interactions with particular wave numbers dictated by the Fermi surface. Taking two-dimensional systems as examples, we find that our effective model captures the underlying physics of the instability toward noncoplanar multiple-$Q$ states discovered recently: a single-$Q$ helical state expected from the RKKY theory is replaced by a double-$Q$ vortex crystal with chirality density waves even for an infinitely small spin-charge coupling on generic lattices [R. Ozawa et al., J. Phys. Soc. Jpn. 85, 103703 (2016)], and a triple-$Q$ Skyrmion crystal with a high topological number of two appears while increasing the spin-charge coupling on a triangular lattice [R. Ozawa, S. Hayami, and Y. Motome, Phys. Rev. Lett. 118, 147205 (2017)]. We find that by introducing an external magnetic field, our effective model exhibits a plethora of multiple-$Q$ states. Our effective model will serve as a guide for exploring further exotic magnetic orderings in itinerant magnets, not only in two dimensions but also in three dimensions.

• Received 22 March 2017
• Revised 26 May 2017

DOI:https://doi.org/10.1103/PhysRevB.95.224424