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\"odinger-type contribution to the Hamiltonian which provides a small perturbation to a dominant Dirac contribution. This changes the Landau level 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 plateaus remains even when the ratio of the Schr\"odinger (E₀) to Dirac (E₁) magnetic energy scale increases either through an increase in B, a decrease in the Schr\"odinger mass, or a decrease of the Dirac Fermi velocity. First corrections to the optical matrix elements (OMEs) in the relativistic case drop out at least to order (E₀/E₁)³. In the opposite limit E₁ small, the quantization remains classical but there is a split into two series. The first corrections to the OMEs in this case cancel out at least to order (E₁/E₀)⁴.