Evolution of the Berry phase and topological properties of a band deformed Chern insulator

Abstract

Here we study the evolution of the Berry phase ($\Phi _{B}$) and the topological properties of a Chern insulator in the presence of a band deformation en route from a Dirac dispersion to an anisotropic Dirac one. Such a scenario can be achieved by tuning one of the hopping amplitudes among a pair of sites in a honeycomb lattice (say, $t_{1}$) with respect to those corresponding to the other two pairs (t). The anisotropic Dirac dispersion characterized by different velocities corresponding to different directions in the k-space is further confirmed by computing the energy (E) dependence of the cyclotron mass which changes from being proportional to $\sqrt{E}$ in the absence of the Haldane flux to linearly in E in its presence. At $t_{1} = 2t$ (known as the semi-Dirac limit), the two Dirac points merge at an intermediate $\mathbf {M}$ point where both the energy gap and the Berry phase vanish in the absence of a Haldane flux, whereas the presence of a flux yields a non-zero Berry phase, even though the spectrum still remains gapless. Moreover, in the absence of the Haldane flux, the Berry phase remains insensitive to the energy of the particle, while in its presence, $\Phi _{B}$ is altered corresponding to different radii of the paths traversed by the particle in the k-space. As $t_{1}$ exceeds a value greater than 2t, a gap re-opens in the energy spectrum, and the Berry phase vanishes even in presence of a Haldane flux indicating the onset of a transition from a topological phase to a trivial one. This transition is further supported by the existence of the edge modes that are present only in the topological regime.