Bloch Electrons in a Uniform Magnetic Field

Abstract

The physical periodicity of a space lattice is not destroyed by the presence of a uniform magnetic field. It is shown that a ray group of unitary operators, isomorphic to pure translations, commutes with the Hamiltonian in this case. Such a group has the characteristic property that $AB=\mathrm{exp}[i\ensuremath{\varphi}(A,B)]C$, where $A$, $B$, and $C$ are elements of the group and $\ensuremath{\varphi}$ is a numerical factor. Representation theory applied to this group yields the characteristic degeneracies of levels in magnetic fields, as well as the transformation properties of eigenfunctions. By means of these it is possible to construct an effective Hamiltonian appropriate to finite magnetic fields in crystals.