Particle-Hole Asymmetry in Gapped Topological Insulator Surface States

We consider the combined effect of a gap and a Zeeman interaction on the helical Dirac fermions which exist on the surface of a topological insulator. Magneto-optical properties, the magnetization, Hall effect and the density of states are considered with emphasis on the particle-hole asymmetry which arises when a subdominant Schr\"odinger piece is included along with the dominant Dirac part of the Hamiltonian. When appropriate, we compare our results with those of a single valley gapped graphene system for which Zeeman splitting behaves differently. We provide a derivation of the phase offset in the magnetic oscillations brought about by the combined effect of the gap and Schr\"odinger term without requiring the semiclassical Onsager quantization condition. Our results agree with previous discussions based on semiclassical arguments.