Landau quantization of graphene including diamagnetic shift and shrinkage of wave function

The electronic band structure and the Landau quantization for graphene in magnetic fields are studied, where we include the diamagnetic shift and shrinkage of wave function by going beyond the effective-mass scheme and the Peierls substitution. This is accomplished with an exactly solvable, honeycomb potential model. In the resulting Landau levels the twofold degeneracy due to the bipartite lattice symmetry is found to be lifted.