Two nonlinear Dicke models

We consider two simple classes of nonlinear extensions of the Dicke model in order to understand how sensitive the presence of the phase transition is to the structural form of the model Hamiltonian. In both classes, second order phase transitions can occur for sufficiently large values of the coupling constant λ. Gap equations are derived for both classes of nonlinear extensions. Second order phase transitions cannot occur in these classes of models unless the model Hamiltonian contains a bilinear interaction term of the form originally proposed by Dicke.