On a mean field approximation for Higgs-Yukawa systems

We discuss the phase structure of a lattice Higgs-Yukawa system in the variational mean field approximation with contributions of fermionic determinant being calculated in a ladder approximation. In particular, we demonstrate that in this approximation the ferrimagnetic phase in the $Z_2$ model with naive fermions can appear as an artifact of a finite lattice and that the phase diagram for this model on infinite lattice changes qualitatively at space-time dimension $D=4$ compared with those at $D>4$.