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We study the effective spin-orbital model for honeycomb-layered transitionmetal compounds, applying the second-order perturbation theory to thethree-orbital Hubbard model with the anisotropic hoppings. This model isreduced to the Kitaev model in the strong spin-orbit coupling limit. Combiningthe cluster mean-field approximations with the exact diagonalization, we treatthe Kugel-Khomskii type superexchange interaction and spin-orbit coupling on anequal footing to discuss ground-state properties. We find that a zigzag orderedstate is realized in the model within nearest-neighbor interactions. We clarifyhow the ordered state competes with the nonmagnetic state, which isadiabatically connected to the quantum spin liquid state realized in a strongspin-orbit coupling limit. Thermodynamic properties are also addressed. Thepresent work should provide another route to account for the Kitaev-basedmagnetic properties in candidate materials.

Role of the spin-orbit coupling in the Kugel-Khomskii model on the honeycomb lattice