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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 the Dzyaloshinskii-Moriya (DM) interaction which drives 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 by including the full set of terms, we identify Heisenberg, Ising-like, and ring exchange interactions, as well as a variety of four-body terms for realistic Hubbard parameters, which impose strong constraints on the relative strengths of the spin interactions. Our spin model shows excellent agreement in energy with the SOC Hubbard model down $U/t \sim 5$ near the metal-insulator transition point, providing insight to which kind of magnetic interactions relevant across this regime are responsible for the emergence of complex magnetic textures.