Anisotropic spin model and multiple-$Q$ states in cubic systems

Multiple-$Q$ states manifest themselves in a variety of noncollinear and noncoplanar magnetic structures depending on the magnetic interactions and lattice structures. In particular, cubic-lattice systems can host a plethora of multiple-$Q$ states, such as magnetic skyrmion and hedgehog lattices. We here classify momentum-dependent anisotropic exchange interactions in the cubic-lattice systems based on the magnetic representation analysis. We construct an effective spin model for centrosymmetric cubic space groups, $Pm\bar{3}m$ and $Pm\bar{3}$, and noncentrosymmetric ones, $P\bar{4}3m$, $P432$, and $P23$: The former include the symmetric anisotropic exchange interaction, while the latter additionally include the Dzyaloshinskii-Moriya interaction. We demonstrate that the anisotropic exchange interaction becomes the origin of the multiple-$Q$ states by applying the anisotropic spin model to the case under $Pm\bar{3}$. We show several multiple-$Q$ instabilities in the ground state by performing simulated annealing. Our results will be a reference for not only exploring unknown multiple-$Q$ states but also understanding the origin of the multiple-$Q$ states observed in both noncentrosymmetric and centrosymmetric magnets like EuPtSi and SrFeO$_3$.