Trigonal warping effects on optical properties of anomalous Hall materials

The topological nature of topological insulators are related to the
symmetries present in the material, for example, 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. We calculate the linear optical conductivity semi-analytically 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 $\mp 1$
instead of $\pm 1/2$. This results in drastic changes in the anomalous Hall and
longitudinal conductivity. The trigonal warping terms also activates the higher
order Hall responses which does not exist in a $\mathcal{R}$ symmetric
conventional Dirac material. We found the presence of a non-zero second order
Hall current even in the absence of Berry curvature dipole. This shift current
is also unaffected by the chirality of the Dirac cone, which should lead to a
non-zero Hall current in time reversal symmetric systems.