Asymptotically exact solution of the generalized Dicke model

The exact solution for the generalized Dicke model is obtained by using a modification of the method of N. N. Bogoliubov (Jr.). Thermodynamics of the boson and spin subsystems is studied. It is shown that the initial system containing interaction with a boson field is thermodynamically equivalent to an anisotropic $X$–$Y$-model with infinitely long range of interaction.