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 α-T₃ 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 (C=1) phase to the higher Chern insulator (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 C=2, indicating that C is insufficient to characterize this system because of the indirect band gap. By plotting the Chern number and C phase diagram, we show that in the presence of both Haldane and modified Haldane models in the α-T₃ lattice, the interplay between the two models manifests three distinct topological phases, namely the C=1 Chern insulator (CI) phase, C=2 higher Chern insulator (HCI) phase, and C=2 topological metal (TM) phase. These results are further supported by the α-T₃ Hall conductance, zigzag, and armchair edge states calculations. This work elucidates the rich phase evolution of the Haldane and modified Haldane models as α varies continuously from 0 to 1 in an α-T₃ model, thus suggesting α-T₃ lattice as a versatile condensed matter platform for studying topological phase transitions.