Numerical study of quantum Hall effect in two-dimensional multi-band system: single- and multi-layer graphene

The Chern numbers which correspond to quantized Hall conductance ${\sigma }_{xy}$ were calculated for single- and bi-layer honeycomb lattices. The quantization of ${\sigma }_{xy}$ occurs in entire energy range. Several large jumps of Chern numbers appear at van-Hove singularities of energy bands without magnetic fields. The plateauxof ${\sigma }_{xy}$ are discussed from semi-classical quantization.