Coupling of the Triple-Q State to the Atomic Lattice by Anisotropic Symmetric Exchange*

The triple-Q (3Q) state has been predicted about 20 years ago as a three-dimensional (non-coplanar) spin structure on a two-dimensional lattice [1]. It exhibits intriguing properties such as topological orbital magnetization, topological charge, and a spontaneous Hall effect. The 3Q state has been first observed in an ultrathin Mn film [2] and very recently also in the layered bulk material Co_1/3TaS₂ confirming the predicted transport properties [3,4]. Here, we identify the ideal 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 [5]. 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, as proposed for Mn/Re(0001) [6], leads to other orientations of the 3Q state which, however, results in a reduced topological orbital magnetization compared to the ideal 3Q state.

[1] Ph. Kurz et al., PRL 86, 1106 (2001)

[2] J. Spethmann et al., PRL124,227203 (2020)

[3] H. Takagi et al., Nat. Phys. 19, 961 (2023)

[4] P. Park et al., arxiv:2303.03760 (2023)

[5] F. Nickel et al., arxiv: 2307.09764 (2023)

[6] S. Haldar et al., Phys. Rev. B 104, L180404 (2021)