Hall conductivity as the topological invariant in magnetic Brillouin zone

Abstract: Hall conductivity for the intrinsic quantum Hall effect in homogeneous systems is given by the topological invariant composed of the Green function depending on momentum of quasiparticle. This expression reveals correspondence with the mathematical notion of the degree of mapping. A more involved situation takes place for the Hall effect in the presence of external magnetic field. In this case the mentioned expression remains valid if the Green function is replaced by its Wigner transformation while ordinary products are replaced by the Moyal products. Such an expression, unfortunately, is much more complicated and might be useless for the practical calculations. Here we represent the alternative representation for the Hall conductivity of a uniform system in the presence of constant magnetic field. The Hall conductivity is expressed through the Green function taken in Harper representation, when its nonhomogeneity is attributed to the matrix structure while functional dependence is on one momentum that belongs to magnetic Brillouin zone. Our results were obtained for the non - interacting systems. But we expect that they remain valid for the interacting systems as well. We, therefore, propose the hypothesis that the obtained expression may be used for the topological description of fractional quantum Hall effect.