Bloch Electrons in Magnetic Fields. Rationality, Irrationality, Degeneracy

The “irrational” equations for determining the energy eigenvalues developed earlier are studied. It is shown that, for rational fields, they can be transformed into the “rational” equations which are generally preferred. The two energy spectra are therefore the same, and there exists no conflict. Further, the question of infinite degeneracy familiar from the Landau functions is taken up. It was established earlier that as long as the field is irrational, this degeneracy is also present for Bloch electrons; now a criterion is developed for the condition under which this degeneracy may be lifted. The criterion comes out to be a rationality condition on the field: it differs in general from the usual (Brown) rationality; it is called “green” rationality.