Virtual processes and super-radiance in spin-boson models

We consider spin-boson models composed by a single bosonic mode and an ensemble of N identical two-level atoms. The situation where the coupling between the bosonic mode and the atoms generates resonant and nonresonant processes is studied, where the whole system is in thermal equilibrium with a reservoir at temperature beta(-1). Phase transitions from ordinary fluorescence to super-radiant phase in three different models are investigated. First a model where the coupling between the bosonic mode and the jth atom is via the pseudo-spin-operator sigma(z)((j)) is studied. Second, we investigate the generalized Dicke model, introducing different coupling constants between the single mode bosonic-field and the environment, g(1) and g(2), for rotating and counter-rotating terms, respectively. Finally a modified version of the generalized Dicke model with intensity-dependent coupling in the rotating terms is considered. In the first model, for an arbitrary value for the coupling constant, the zero mode contributes in rendering the canonical entropy a negative quantity for low temperatures. For small coupling constants, the partition function is analytic for all temperatures. The last two models present phase transitions, even when only Hamiltonian terms which generate virtual processes are considered.