Formation mechanism of helical Q structure in Gd-based skyrmion materials

Using the ab initio local force method, we investigate the formation mechanism of the helical spin structure in GdRu$_2$Si$_2$ and Gd$_2$PdSi$_3$. We calculate the paramagnetic spin susceptibility and find that the Fermi surface nesting is not the origin of the incommensurate modulation, in contrast to the naive scenario based on the Ruderman-Kittel-Kasuya-Yosida mechanism. We then decompose the exchange interactions between the Gd spins into each orbital component, and show that spin-density-wave type interaction between the Gd-5$d$ orbitals is ferromagnetic, but the interaction between the Gd-4$f$ orbitals is antiferromagnetic. We conclude that the competition of these two interactions, namely, the inter-orbital frustration, stabilizes the finite-Q structure.