ENERGY OF ELECTRONS ON A 2-DIMENSIONAL LATTICE IN A MAGNETIC-FIELD - PERTURBATION-THEORY VERSUS PEIERLS SUBSTITUTION

Energy levels and total energy of electrons in a magnetic field moving in a two dimensional periodic potential are obtained. Two limiting cases are treated: the tight binding limit and the weak potential. In these cases different direct perturbation theories are applicable. The energy difference with and without magnetic field is compared with calculations based on the "Peierls substitution". The latter is shown to lead to severe qualitative and quantitative errors. The sources of these errors are clarified. The spontaneous generation of a magnetic field is shown to be impossible.