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Ferromagnetism in the two-dimensional CrI$_3$ has generated a lot ofexcitement, and it was recently proposed that the spin-orbit coupling (SOC) inIodine may generate bond-dependent spin interactions leading to magneticanisotropy. Here we derive a microscopic spin model of S=3/2 on transitionmetals surrounded by heavy ligands in honeycomb Mott insulators using astrong-coupling perturbation theory. For ideal octahedra we find Heisenberg andKitaev interactions, which favor the magnetic moment along the cubic axis viaquantum fluctuations. When a slight trigonal distortion of the octahedra ispresent together with the SOC, three additional interactions arise, comprisedof the off-diagonal symmetric $\Gamma$ and $\Gamma^\prime$, and single-ionanisotropy. The resulting magnetic anisotropy pins the moment perpendicular tothe honeycomb plane as observed in a single-layer of CrI$_3$, suggesting thesignificance of SOC and trigonal distortion in understanding magnetism of twodimensional Mott insulators. Comparison to the spin-orbit coupled $J_{\rmeff}$= 1/2 and S=1 models is also presented.

Magnetic Anisotropy in Spin-3/2 with Heavy Ligand in Honeycomb Mott Insulators: Application to CrI$_3$