  Conductivity of Integer Quantum Hall Effect (IQHE) may be expressed as the
topological invariant composed of the two - point Green function. Such a
topological invariant is known both for the case of homogeneous systems with
intrinsic Anomalous Quantum Hall Effect (AQHE) and for the case of IQHE in the
inhomogeneous systems. In the latter case we may speak, for example, of the
AQHE in the presence of elastic deformations and of the IQHE in presence of
magnetic field. The topological invariant for the general case of inhomogeneous
systems is expressed through the Wigner transformed Green functions and
contains Moyal product. When it is reduced to the expression for the IQHE in
the homogeneous systems the Moyal product is reduced to the ordinary one while
the Wigner transformed Green function (defined in phase space) is reduced to
the Green function in momentum space. Originally the mentioned above
topological representation has been derived for the non - interacting systems.
We demonstrate that in a wide range of different cases in the presence of
interactions the Hall conductivity is given by the same expression, in which
the noninteracting two - point Green function is substituted by the complete
two - point Green function with the interactions taken into account. Several
types of interactions are considered including the contact four - fermion
interactions, Yukawa and Coulomb interactions. We present the complete proof of
this statement up to the two loops, and argue that the similar result remains
to all orders of perturbation theory. It is based on the incorporation of
Wigner - Weyl calculus to the perturbation theory. We, therefore, formulate
Feynmann rules of diagram technique in terms of the Wigner transformed
propagators.
