Interplay between Haldane and modified Haldane models in α−T3 lattice: Band structures, phase diagrams, and edge states

We study the topological properties of the Haldane and modified Haldane models in $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ lattice. The band structures and phase diagrams of the system are investigated. Individually, each model undergoes a distinct phase transition: (i) the Haldane-only model experiences a topological phase transition from the Chern insulator $(\mathcal{C}=1)$ phase to the higher Chern insulator $(\mathcal{C}=2)$ phase; while (ii) the modified-Haldane-only model experiences a phase transition from the topological metal phase to the higher Chern insulator phase with identical Chern number $\mathcal{C}=2$, indicating that $\mathcal{C}$ is insufficient to characterize this system because of the indirect band gap. By plotting the Chern number and $\mathcal{C}$ phase diagram, we show that in the presence of both Haldane and modified Haldane models in the $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ lattice, the interplay between the two models manifests three distinct topological 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 $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ Hall conductance, zigzag, and armchair edge states calculations. This work elucidates the rich phase evolution of the Haldane and modified Haldane models as $\ensuremath{\alpha}$ varies continuously from 0 to 1 in an $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ model, thus suggesting $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ lattice as a versatile condensed matter platform for studying topological phase transitions.