Magnetic anisotropy in spin-3/2 with heavy ligand in honeycomb Mott insulators: Application to CrI_{3}

Ferromagnetism in 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, composed of the off-diagonal symmetric Γ and Γ^{′} 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. A comparison with the spin-orbit coupled J_{eff}=1/2 and S=1 models is also presented.