Seminar: Systemic Band Deformation and Topological Phase Transition in Honeycomb Lattice

We study the evolution of the topological properties of a bilayer Chern insulator and a quantum spin Hall insulator subjected to band engineering. The manipulation of band deformities, achieved by introducing an anisotropic hopping parameter, denoted as t1, along one of the three neighboring sites compared to the other (denoted by t), leads to profound topological phase transitions. Additionally, the incorporation of Haldane’s second neighbor hopping, disrupting time-reversal symmetry and opening a gap in the dispersion spectrum, adds complexity to the system. Under the band deformation, the band structure of Bernal stacked bilayer system demonstrates vanishing of the band gap at t1 = 2t. The nature of the gap on either side of this threshold, marked by the continuous absence of time-reversal symmetry, is distinct, as elucidated through the calculation of Chern numbers, which vanish beyond t1 = 2t. Furthermore, the band closer to the Fermi level exhibits higher Chern numbers, such as, C = ±2, while bands farther away possess only C = ±1. These contrasting Chern numbers form a phase diagram, providing insight into multiple phase transitions. These transitions are substantiated by the presence or absence of chiral edge modes in semi-infinite nanoribbons and the emergence of quantized plateaus in the Hall conductivity, offering a comprehensive view of the dynamic behavior of the system. Moreover, the corresponding scenario in a quantum spin Hall insulator described by a Kane-Mele model shows the spin-resolved bands. They respond similarly to the band deformation, as evidenced by the computation of the Z2 invariant. The evolution of the spin Hall response uncovers the vanishing of the quantum spin hall phase at t1 = 2t, further enriching our understanding of the system’s intricate behavior.