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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$_3$ andits isostructural materials in nature have been turned out to be a Mottinsulator with magnetic order. Here we theoretically revisit the topologicalproperty based on a minimal tight-binding Hamiltonian for three $t_{2g}$ bandsincorporating a strong spin orbit coupling and two types of the first nearestneighbor (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 topologicalphase transition takes place by varying only these hopping parameters with therelative 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 NNhopping channels, and the trigonal distortion on the topological phase for thewhole range of $\theta$. Furthermore, we examine the electronic and topologicalphases in the presence of on-site Coulomb repulsion $U$. Employing the clusterperturbation theory, we show that, with increasing $U$, a trivial ortopological band insulator in the absence of $U$ can be transferred into a Mottinsulator with nontrivial or trivial band topology. We also show that the maineffect of the Hund's coupling can be understood simply as the renormalizationof $U$. We briefly discuss the relevance of our results to the existingmaterials.

Topological property of a $t_{2g}^5$ system with a honeycomb lattice structure