Thermal phase transition for some spin-boson models

•We study the thermodynamics of the Dicke model considering spatial separation between atoms.•We study the thermodynamics of the Thompson model which considers phonons in the Dicke model.•Using the path integral approach, we compute the free energy and the collective spectrum of the models.•We identify a Goldstone mode when the continuous symmetry is spontaneously broken in both models.

In this work we study two different spin-boson models. Such models are generalizations of the Dicke model, it means they describe systems of NN identical two-level atoms coupled to a single-mode quantized bosonic field, assuming the rotating wave approximation. In the first model, we consider the wavelength of the bosonic field to be of the order of the linear dimension of the material composed of the atoms, therefore we consider the spatial sinusoidal form of the bosonic field. The second model is the Thompson model, where we consider the presence of phonons in the material composed of the atoms. We study finite temperature properties of the models using the path integral approach and functional methods. In the thermodynamic limit, N→∞N→∞, the systems exhibit phase transitions from normal to superradiant phase at some critical values of temperature and coupling constant. We find the asymptotic behavior of the partition functions and the collective spectrums of the systems in the normal and the superradiant phases. We observe that the collective spectrums have zero energy values in the superradiant phases, corresponding to the Goldstone mode associated to the continuous symmetry breaking of the models. Our analysis and results are valid in the limit of zero temperature β→∞β→∞, where the models exhibit quantum phase transitions.