Hall conductivity as topological invariant in phase space

It is well known that quantum Hall conductivity in the presence of a constant magnetic field is expressed through the topological TKNN invariant. The same invariant is responsible for the intrinsic anomalous quantum Hall effect. We propose a generalization of these expressions to the QHE in the presence of non-uniform external fields. Our approach is based on purposely developed lattice Wigner-Weyl formalism, giving the Hall conductivity in terms of Weyl symbols of the two-point Green's function. It is shown to be topological invariant in the phase space of the system.