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We study the topological properties of the Haldane and modified Haldanemodels in $\alpha$-$T_{3}$ lattice. The band structures and phase diagrams ofthe system are investigated. Individually, each model undergoes a distinctphase transition: (i) the Haldane-only model experiences a topological phasetransition from the Chern insulator ($\mathcal{C} = 1$) phase to the higherChern insulator ($\mathcal{C} = 2$) phase; while (ii) the modified-Haldane-onlymodel experiences a phase transition from the topological metal ($\mathcal{C} =2$) phase to the higher Chern insulator ($\mathcal{C} = 2$) phase and we showthat $\mathcal{C}$ is insufficient to characterize this system because$\mathcal{C}$ remains unchanged before and after the phase transition. Byplotting the Chern number and $\mathcal{C}$ phase diagram, we show that in thepresence of both Haldane and modified Haldane models in the $\alpha$-$T_{3}$lattice, the interplay between the two models manifests three distincttopological phases, namely the $\mathcal{C} = 1$ Chern insulator (CI) phase,$\mathcal{C} = 2$ higher Chern insulator (HCI) phase and $\mathcal{C} = 2$topological metal (TM) phase. These results are further supported by the$\alpha$-$T_{3}$ zigzag edge states calculations. Our work elucidates the richphase evolution of Haldane and modified Haldane models as $\alpha$ variescontinuously from $0$ to $1$ in an $\alpha$-$T_3$ model.

Interplay between Haldane and modified Haldane models in $\alpha$-$T_{3}$ lattice: Band structures, phase diagrams and edge states