Figure 2: Multi-scale modelling of Cu₂OSeO₃.

(a) The crystal structure is shaped by Cu(2)O₄ plaquettes (yellow) and Cu(1)O₅ bipyramids (orange), and covalent Se-O bonds (thick lines), forming a sparse three-dimensional lattice. This lattice can be tiled into tetrahedra (dashed lines), each composed of one Cu(1) and three Cu(2) sites, depicted by large light brown and light cyan spheres, respectively. (Smaller spheres show the remaining non-magnetic sites). (b) The magnetic Cu²⁺ ions form a distorted pyrochlore lattice, a network of corner-shared tetrahedra. DFT calculations evidence the presence of both types of magnetic interactions—antiferromagnetic (red) and ferromagnetic (blue), in agreement with experimental magnetic structure (arrows). The strength of a certain coupling is indicated by the thickness of the respective line. The strongest couplings, and , are found within the tetrahedra (shaded), while the couplings between the tetrahedra, , , and (the latter is a longer-range coupling), are substantially weaker (dashed lines). (c) The quantum-mechanical treatment of a single tetrahedron t yields a magnetic spin S_t=1 ground state, separated from the lowest lying excitation by a large gap Δ≃275 K. (d) The tetrahedra reside at the vertices of a trillium lattice, exactly like the Mn ions in MnSi. The quantum-mechanical nature of the effective moments is indicated by sectioned arrows.