Trigonal warping effects on optical properties of anomalous Hall materials

The topological nature of topological insulators is related to the symmetries present in the material, for example, the quantum spin Hall effect can be observed in topological insulators with time-reversal symmetry, while broken time-reversal symmetry may give rise to the presence of anomalous quantum Hall effect (AHE). Here we consider the effects of broken rotational symmetry on the Dirac cone of an AHE material by adding trigonal warping terms to the Dirac Hamiltonian. By calculating the linear optical conductivity semi-analytically, we can show that the Hall conductivity from each Dirac cone is quantized with Chern number ±1. The ±1 Chern number is caused by the appearance of additional Dirac cones due to the trigonal warping terms. This results in drastic changes in the anomalous Hall and longitudinal conductivity. The trigonal warping terms also activate the higher-order Hall responses which do not exist in an R-symmetric conventional Dirac material. We found the presence of a non-zero second-order transversal current contribution from the shift current even in the absence of Berry curvature dipole. This shift current does not vanish due to contributions of opposite chiralities rendering it observable in time-reversal symmetric systems.