  The anomalous quantum Hall conductivity in the 2+1D topological insulators in
the absence of interactions may be expressed as the topological invariant
composed of the two - point Green function. For the noninteracting system this
expression is the alternative way to represent the TKNN invariant. It is widely
believed that 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. However, the proof of this statement has not been given so
far. In the present paper we give such a proof in the framework of the
particular tight - binding models of the $2+1$ D topological insulator.
Besides, we extend our consideration to the $3+1$ D Weyl semimetals. It was
known previously that with the interactions neglected the Hall conductivity in
those systems is expressed through the two - point Green function in the way
similar to that of the $2+1$ D topological insulators. Again, the influence of
interactions on this expression has not been investigated previously. We
consider this problem within the framework of the particular $3+1$D model of
Weyl semimetal in the presence of the contact four - fermion interactions and
Coulomb interactions. We prove (up to the one - loop approximation), that the
Hall conductivity is given by the same expression as in the noninteracting
case, in which the noninteracting Green function is substituted by the complete
two - point Green function with the interactions included. Basing on the
obtained expressions we discuss the topological phase transitions accompanied
by the change of Hall conductivity.
