Coupling of the triple- q state to the atomic lattice by anisotropic symmetric exchange

We identify the triple-$\mathbf{q}$ (3Q) state as magnetic ground state in Pd/Mn and Rh/Mn bilayers on Re(0001) using spin-polarized scanning tunneling microscopy and density-functional theory. An atomistic model reveals that in general the 3Q state with tetrahedral magnetic order and zero net spin moment is coupled to a hexagonal atomic lattice in a highly symmetric orientation via the anisotropic symmetric exchange interaction, whereas other spin-orbit coupling terms cancel due to symmetry. Our experiments are in agreement with the predicted orientation of the 3Q state. A distortion from the ideal tetrahedral angles would lead to other orientations of the 3Q state with a reduced topological orbital magnetization compared to the ideal 3Q state.