From hierarchical triangular spin liquid to multi-$q$ spin texture in spinel GeFe$_2$O$_4$

Combining macroscopic measurements, neutron scattering and modeling, we identify in the GeFe$_2$O$_4$ spinel a correlated paramagnetic state resulting from the predominance of third-neighbor antiferromagnetic interactions. These interactions materialize 4 isolated families of triangular planes with 120$^{\circ}$ spins emerging from the underlying pyrochlore lattice. At lower temperatures, a phase transition occurs from this hierarchical spin liquid to a non-coplanar spin texture that is characterized by 6 propagating vectors. This unusual multi-$q$ order is triggered by the presence of weaker interactions up to the sixth neighbors. The system is remarkably successful in coupling the different triangular planes while maintaining their two-dimensional 120$^{\circ}$ order. Our study highlights the hierarchy of interactions involved in GeFe$_2$O$_4$, which is singular among spinel compounds since first-neighbor interactions are only a small fraction of the dominant third neighbor ones.