Spin-orbital state induced by strong spin-orbit coupling

To clarify a crucial role of a spin-orbit coupling in the emergence of novel spin-orbital states in $5d$-electron compounds such as Sr${}_2$IrO${}_4$, we investigate ground state properties of a $t_{2\mathrm{g}}$-orbital Hubbard model on a square lattice by Lanczos diagonalization. In the absence of the spin-orbit coupling, the ground state is spin singlet. When the spin-orbit coupling is strong enough, the ground state turns into a weak ferromagnetic state. The weak ferromagnetic state is a singlet state in terms of an effective total angular momentum. Regarding the orbital state, we find the so-called complex orbital state, in which real $xy$, $yz$, and $zx$ orbital states are mixed with complex coefficients.