  The Dicke model describes an ensemble of N identical two-level atoms (qubits)
coupled to a single mode of a bosonic field. The fermion Dicke model should be
obtained by changing the atomic pseudo-spin operators by a linear combination
of Fermi operators. The generalized fermion Dicke model is defined introducing
different coupling constants between the single mode of the bosonic field and
the reservoir. In the thermodynamic limit, the fermion Dicke model can be
analized using the path integral approach with functional method. The system
exhibits a second order phase transition from normal to superrandiance at some
critical temperature with the presence of a condensate. We evaluate the
critical transition temperature and present the spectrum of the collective
bosonic excitations. There is quantum phantum critical behavior when the
coupling constants satisfy an especific condition. Two particular situations
are analyzed. First, we present the spectrum of the collective bosonic
excitations in the case using the rotating-wave approximation, recovering the
well know results. Second, the case only considering virtual processes. In the
last case, it is possible to have a superradiance phase when only virtual
processes are introduced in the interaction Hamiltonian. Here also appears a
quantum phase transition at some critical coupling, and for larger values for
the critical coupling, the system enter in this superradiant phase with a
Goldstone mode.
