New topological invariant for the problem of quantum hall effect in a two-dimensional periodic potential

The special case of the two magnetic subbands arising from a degenerate wavefunctions of Landau level when the electrons are subjected to a weak periodic potential as well as a strong magnetic field is studied in detail. The condition for the existence of the gap between the bands is stated. It is shown that the wavefunction of an electron regarded as function of wavevector k has the peculiar singularities, i.e. the branch points of its phase. The Kubo formula for the Hall conductance is written in the form of the new topological invariant that is a degree of Gaussian mapping of the torus into the unit sphere. The invariant is calculated for some quite general example.