Coupling effect of topological states and Chern insulators in two-dimensional triangular lattices

We investigate topological states of two-dimensional (2D) triangular lattices with multi-orbitals. Tight-binding model calculations of a 2D triangular lattice based on ${\mathit{\text{p}}}_x$ and \emph{p}_{y} orbitals exhibit very interesting doubly degenerate energy points at different positions ($\mathrm{\Gamma }$ and K/K${}^{\mathrm{\prime }}$) in momentum space, with quadratic non-Dirac and linear Dirac band dispersions, respectively. Counterintuitively, the system shows a global topologically trivial rather than nontrivial state with consideration of spin-orbit coupling due to the “destructive interference effect” between the topological states at the $\mathrm{\Gamma }$ and K/K${}^{\mathrm{\prime }}$ points. The topologically nontrivial state can emerge by introducing another set of triangular lattices to the system (bitriangular lattices) due to the breakdown of the interference effect. With first-principles calculations, we predict an intrinsic Chern insulating behavior (quantum anomalous Hall effect) in a family of 2D triangular lattice metal-organic framework of Co(C${}_{21}$N${}_3$H${}_{15}$) (TPyB-Co) from this scheme. Our results provide a different path and theoretical guidance for the search for and design of new 2D topological quantum materials.