The exact diagonalization and the variational cluster approximation (VCA) are used to study the nature of a novel Mott insulator induced by a strong spin-orbit coupling for a two-dimensional three-band Hubbard model consisting of the $t_{2g}$ manifold of $5d$ orbitals. To characterize the ground state, we introduce a local Kramers doublet which can represent a state with effective angular momentum $J_{rm eff}=|{bm S}-{bm L}|=1/2$ as well as spin $S=1/2$. Our systematic study of the pseudo-spin structure factor defined by the Kramers doublet shows that the $J_{rm eff}=1/2$ Mott insulator is smoothly connected to the $S=1/2$ Mott insulator. Using the Kramers doublet as a variational state for the VCA, we examine the one-particle excitations for the Mott insulating phase. These results are compared with recent experiments on Sr$_2$IrO$_4$.
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