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

## TL;DR

This paper investigates the topological phases of a $t_{2g}^5$ honeycomb lattice system, analyzing how hopping parameters, distortions, and electron interactions influence topological and Mott insulating states.

## Contribution

It provides a theoretical framework showing how varying hopping parameters and Coulomb interactions induce topological phase transitions in $t_{2g}^5$ systems.

## Key findings

- Topological phase transition driven by hopping parameter variation.
- Coulomb interaction can induce transitions between trivial and nontrivial insulators.
- Hund's coupling effectively renormalizes the Coulomb interaction.

## Abstract

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.