  A $t_{2g}^5$ system with a honeycomb lattice structure such as Na$_2$IrO$_3$
was firstly proposed as a topological insulator even though Na$_2$IrO$_3$ and
its isostructural materials in nature have been turned out to be a Mott
insulator with magnetic order. Here we theoretically revisit the topological
property based on a minimal tight-binding Hamiltonian for three $t_{2g}$ bands
incorporating a strong spin orbit coupling and two types of the first nearest
neighbor (NN) hopping channel between transition metal ions, i.e., the hopping
($t_1$) mediated by edge-shared ligands and the direct hopping ($t_1'$) between
$t_{2g}$ orbitals via $dd\sigma$ bonding. We demonstrate that the topological
phase transition takes place by varying only these hopping parameters with the
relative strength parametrized by $\theta$, i.e., $t_1=t\cos\theta$ and
$t_1'=t\sin\theta$. We also explore the effect of the second and third NN
hopping channels, and the trigonal distortion on the topological phase for the
whole range of $\theta$. Furthermore, we examine the electronic and topological
phases in the presence of on-site Coulomb repulsion $U$. Employing the cluster
perturbation theory, we show that, with increasing $U$, a trivial or
topological band insulator in the absence of $U$ can be transferred into a Mott
insulator with nontrivial or trivial band topology. We also show that the main
effect of the Hund's coupling can be understood simply as the renormalization
of $U$. We briefly discuss the relevance of our results to the existing
materials.
