Manifestation of on-site Coulomb and spin-orbit interactions in the ground state electronic structure of Sr$_2$IrO$_4$

In contrast to the superconducting and metallic ground states in Sr$_2$RuO$_4$ and Sr$_2$RhO$_4$, the ground state of Sr$_2 $IrO$_4$ has been reported to be a magnetic insulator. Such an insulating character of Sr$_2$IrO$_4$ is rather surprising and unexpected when the extended nature of Ir 5$d$ state is considered. To investigate the electronic structure of Sr$_2 $IrO$_4$, we performed LDA+U calculations taking account of spin-orbit interactions, where both on-site Coulomb interactions and spin-orbit couplings in the description of Ir 5 $d$ states are expected to play a significant role. From the results, it is shown that neither the on-site U nor the spin- orbit term only can explain the insulating feature of Sr$_2 $IrO$_4$. An interesting interplay between the two competing interactions is found to determine the spin and orbital configuration, leading to a novel insulating ground state. To understand the nature of the ground state, we suggest a minimal model for the $t_{2g}$ manifold based on the tight binding Hamiltonian.