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

Ferromagnetism in the two-dimensional CrI$_3$ has generated a lot of excitement, and it was recently proposed that the spin-orbit coupling (SOC) in Iodine may generate bond-dependent spin interactions leading to magnetic anisotropy. Here we derive a microscopic spin model of S=3/2 on transition metals surrounded by heavy ligands in honeycomb Mott insulators using a strong-coupling perturbation theory. For ideal octahedra we find Heisenberg and Kitaev interactions, which favor the magnetic moment along the cubic axis via quantum fluctuations. When a slight trigonal distortion of the octahedra is present together with the SOC, three additional interactions arise, comprised of the off-diagonal symmetric $\Gamma$ and $\Gamma^\prime$, and single-ion anisotropy. The resulting magnetic anisotropy pins the moment perpendicular to the honeycomb plane as observed in a single-layer of CrI$_3$, suggesting the significance of SOC and trigonal distortion in understanding magnetism of two dimensional Mott insulators. Comparison to the spin-orbit coupled $J_{\rm eff}$= 1/2 and S=1 models is also presented.