  We consider the full Dicke spin-boson model composed by a single bosonic mode
and an ensemble of $N$ identical two-level atoms with different couplings for
the resonant and anti-resonant interaction terms, and incorporate a
dipole-dipole interaction between the atoms. Assuming that the system is in
thermal equilibrium with a reservoir at temperature $\beta^{-1}$, we compute
the free energy in the thermodynamic limit $N\rightarrow\infty$ in the
saddle-point approximation to the path integral and determine the critical
temperature for the superradiant phase transition. In the zero temperature
limit, we recover the critical coupling of the quantum phase transition,
presented in the literature.
