Vortex Crystal Stabilized by the Competition Between Multi-Spin and Out-of-Plane Dzyaloshinskii–Moriya Interactions

Multiple-Q magnetic states encompass a broad class of noncollinear and noncoplanar spin textures generated by the superposition of spin density waves. In this study, we theoretically explore the emergence of vortex crystals formed by multiple-Q spin density waves on a two-dimensional triangular lattice with D3h point group symmetry. Using simulated annealing applied to an effective spin model, we demonstrate that the synergy among the easy-plane single-ion anisotropy, the biquadratic interaction, and the out-of-plane Dzyaloshinsky–Moriya interaction defined in momentum space can give rise to a variety of double-Q and triple-Q vortex crystals. We further examine the role of easy-plane single-ion anisotropy in triple-Q vortex crystals and show that weakening the anisotropy drives topological transitions into skyrmion crystals with skyrmion numbers ±1 and ±2. The influence of an external magnetic field is also analyzed, revealing a field-induced phase transition from vortex crystals to single-Q conical spirals. These findings highlight the crucial role of out-of-plane Dzyaloshinskii–Moriya interactions in stabilizing unconventional vortex crystals, which cannot be realized in systems with purely polar or chiral symmetries.