Superradiant phase transitions of Fermi and bose operators with trilinear interaction

A model of the interaction of a field Bose operator with $2N$ Fermi (or Bose) lattice oscillators is considered. It is shown that in the limit $N\to\infty$ an equilibrium superradiant phase transition of the system can occur in the case of a Bose lattice for arbitrarily small (in contrast to a Fermi lattice) values of the coupling constant and the reciprocal temperature. The thermodynamics of the model is investigated. The results are interpreted and compared with the ordering in a Bose lattice at finite $N$.