Particle-hole asymmetry on Hall conductivity of a topological insulator

The helical Dirac states on the surface of a topological insulator are protected by topology and display significant particle-hole asymmetry. This asymmetry arises from a subdominant Schr\"{o}dinger type contribution to the Hamiltonian which provides a small perturbation to a dominant Dirac contribution. This changes the Landau levels energies in an external magnetic field ($B$) and provides modifications to the usual relativistic optical matrix elements. Nevertheless we find that the relativistic quantization of the Hall plateaux remains even when the ratio of the Schr\"{o}dinger ($E_0$) to Dirac ($E_1$) magnetic energy scale increases either through an increase in $B$, a decrease in the Schr\"{o}dinger mass or of the Dirac fermi velocity. First corrections to the optical matrix elements(OME) in the relativistic case drop out at least to order $(E_0/E_1)^3$. In the opposite limit $E_1$ small, the quantization remains classical but there is a split into two series. The first corrections to the OME in this case, cancel out at least to order $(E_1/E_0)^4$.