Path integral approach to the full Dicke model

TL;DR

This paper analyzes the full Dicke model at finite temperature using path integral methods, identifying phase transitions and symmetry breaking in different coupling regimes, and deriving the system's asymptotic behavior and spectrum.

Contribution

It provides a comprehensive path integral analysis of the full Dicke model, including finite temperature effects and symmetry considerations, which was not previously fully explored.

Findings

Identifies phase transition points at critical temperature and coupling constants.

Derives the asymptotic behavior of the partition function in the thermodynamic limit.

Finds the collective spectrum and Goldstone modes in different symmetry cases.

Abstract

The full Dicke model describes a system of $N$ identical two level-atoms coupled to a single-mode quantized bosonic field. The model considers rotating and counter-rotating coupling terms between the atoms and the bosonic field, with coupling constants $g_1$ and $g_2$, for each one of the coupling terms, respectively. We study finite temperature properties of the model using the path integral approach and functional methods. In the thermodynamic limit, $N\rightarrow\infty$, the system exhibits phase transition from normal to superradiant phase, at some critical values of temperature and coupling constants. We distinguish between three particular cases, the first one corresponds to the case of rotating wave approximation, which $g_1\neq 0$ and $g_2=0$, the second one corresponds to the case of $g_1=0$ and $g_2\neq 0$, in these two cases the model has a continuous symmetry. The last one, corresponds to the case of $g_1\neq 0$ and $g_2\neq 0$, which the model has a discrete symmetry. The phase transition in each case is related to the spontaneous breaking of its respective symmetry. For each one of these three particular cases, we find the asymptotic behaviour of the partition function in the thermodynamic limit, and the collective spectrum of the system in the normal and the superradiat phase. For the case of rotating wave approximation, and also the case of $g_1=0$ and $g_2\neq 0$, in the superradiant phase, the collective spectrum has a zero energy value, corresponding to the Goldstone mode associated to the continuous symmetry breaking of the model. Our analyse and results are valid in the limit of zero temperature, $\beta\rightarrow\infty$, in which, the model exhibits a quantum phase transition.