Vortex order in magnetic frustrated GeNi${}_2$O${}_4$ and GeCo${}_2$O${}_4$ spinels

In the search for new spin textures based on singular magnetic objects like Bloch-points or vortices, spinel compounds emerge as an interesting playground due to the interplay between magnetic anisotropy and complex interactions that extend well beyond first neighbors on a pyrochlore lattice. Based on an exploration of the exchange interaction phase diagrams of members of the Ge$B_2$O${}_4$ family with $B$=Co and Ni, we show, using simultaneous modeling of inelastic neutron scattering measurements and single-crystal neutron diffraction data, that a 2-$k$ magnetic structure may be stabilized in these compounds. This leads to a short period spin vortex crystal, a variant induced by the magnetic anisotropy of the 3-$k$ Bloch-point structure predicted for isotropic spins. Our study rationalizes the formation of these multi-$k$ spin textures in frustrated antiferromagnets, as well as their anisotropy-dependent evolution.