Theoretical study of phase transition from normal to topological insulators in Na$_2$IrO$_3$

Recently Na$_2$IrO$_3$ has been suggested to have possible quantum spin Hall effect arising from the novel $j_{\rm eff}=1/2$ state of 5$d$ Ir atoms. The electronic structure of the layered iridium oxides with honeycomb lattice is investigated based on a tight-binding model with spin-orbit coupling included. Our tight-binding model, fitted to the first-principles calculation results, reveals that the electronic states near the Fermi level are not the $j_{\rm eff}=1/2$ states but the $e'_g$ states. The delocalized 5$d$ orbitals lying in the edge sharing octahedron structure leads to (i) a significant direct hopping between neighboring Ir 5$d$ states, (ii) a strong trigonal crystal field, and (iii) non-negligible next-nearest-neighbor and next-next-nearest-neighbor hoppings. A peculiar band structure is found to play a crucial role in determination of the topological nature of the spin-orbit coupled ground state in Na$_2$IrO$_3$.