Effective spin model with anisotropic exchange interactions for the spin-orbit coupled Hubbard model at half-filling

Spin-orbit coupling (SOC) in noncentrosymmetric materials is the source of incommensurate magnetic structures. In semiconductors, it drives the Rashba spin splitting and spin momentum locking, while in magnetic insulators based on transition metals, it induces anisotropic spin exchange interactions, like Dzyaloshinskii-Moriya (DM) interaction which drive chiral magnetism and skyrmion formation. Here, we establish a direct connection between SOC and spin exchange interactions by deriving an effective spin model from the SOC Hubbard model at half-filling. Using a strong-coupling expansion up to fourth order, we identify Heisenberg, Ising-like, and ring exchange interactions, as well as a variety of four-body terms for realistic Hubbard parameters. These parameters constrain the relative strengths of spin interactions, providing a natural interpolation between metallic and insulating phases that host complex magnetic textures.