  We present an anti-ferromagnetically ordered ground state of
  Na$_{2}$IrO$_{3}$ based on density-functional-theory calculations including
both spin-orbit coupling and on-site Coulomb interaction $U$. We show that the
splitting of $e_{g}'$ doublet states by the strong spin-orbit coupling is
mainly responsible for the intriguing nature of its insulating gap and magnetic
ground state. Due to its proximity to the spin-orbit insulator phase, the
magnetic ordering as obtained with finite $U$ is found to exhibit a strong
in-plane anisotropy. The phase diagram of Na$_{2}$IrO$_{3}$ suggests a possible
interplay between spin-orbit insulator and Mott anti-ferromagnetic insulator
phases.

Strongly Correlated Electrons;Materials Science