Quartic asymmetric exchange for two-dimensional ferromagnets with trigonal prismatic symmetry

We suggest a possible origin of noncollinear magnetic textures in ferromagnets (FMs) with the ${D}_3h$ point group symmetry. The suggested mechanism is different from the Dzyaloshinskii-Moriya interaction (DMI) and its straightforward generalizations. The considered symmetry class is important because a large fraction of all single-layer intrinsic FMs should belong to it, including monolayer ${\mathrm{Fe}}₃\mathrm{Ge}{\mathrm{Te}}₂$. At the same time, the DMI vanishes identically in materials described by this point group, in the continuous limit. We use symmetry analysis to identify the only possible contribution to the free energy density in two dimensions that is of the fourth order with respect to the local magnetization direction and linear with respect to its spatial derivatives. This contribution predicts long-range conical magnetic spirals with both the average magnetization and the "average chirality" dependent on the spiral propagation direction. We relate the predicted spirals to a recent experiment on ${\mathrm{Fe}}₃\mathrm{Ge}{\mathrm{Te}}₂$. Finally, we demonstrate that, for easy-plane materials, the same mechanism may stabilize bimerons.