Multiple- ${Q}$ Magnetic States in Spin–Orbit Coupled Metals

The interplay among charge, spin, and orbital degrees of freedom in electrons gives rise to a variety of electronic states and topological properties. We here report our theoretical study of the instability toward multiple-Q magnetic ordering with noncollinear and noncoplanar spin textures in spin-orbit coupled metals. By simulated annealing for an effective spin model, which incorporates the spin-charge coupling and the Rashba spin-orbit coupling, we elucidate the phase diagram while changing the anisotropic exchange interactions, the antisymmetric Dzyaloshinskii-Moriya interaction, and the external magnetic field. We show that the model exhibits a plethora of multiple-Q states including Néel-type and Bloch-type vortex crystals. Our systematic analysis will provide a guide for complex spin textures observed in spin-orbit coupled metals, such as monolayer metals on substrates.