Trigonal warping effects on optical properties of anomalous Hall materials

The topological nature of solids is related to the symmetries present in the material, for example, a 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 an 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. We calculate the linear optical conductivity semianalytically to show how by breaking the rotational symmetry we can obtain a topologically distinct phase. The addition of trigonal warping terms causes the emergence of additional Dirac cones, which when combined has a total Chern number of ∓1 instead of ±1/2. 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 a ℛ-symmetric conventional Dirac material. We found the presence of a nonzero second-order Hall current even in the absence of a Berry curvature dipole. This shift current is also unaffected by the chirality of the Dirac cone, which should lead to a nonzero Hall current in time-reversal symmetric systems.