Universality class and exact phase boundary in the superradiant phase transition

The Dicke model and Rabi model can undergo phase transitions from the normal phase to the superradiant phase at the same boundary, which can be accurately determined using some approximated approaches. The underlying mechanism for this coincidence is still unclear and the universality class of these two models is elusive. Here we prove this phase transition exactly using the path-integral approach based on the faithful Schwinger fermion representation, and give a unified phase boundary condition for these models. We demonstrate that at the phase boundary, the fluctuation of the bosonic field is vanished, thus it can be treated as a classical field, based on which a much simplified method to determine the phase boundary is developed. This explains why the approximated theories by treating the operators as classical variables can yield the exact boundary. We use this method to study several similar spin and boson models, showing its much wider applicability than the previously used approaches. Our results demonstrate that these phase transitions belong to the same universality by the classical Landau theory of phase transitions, which can be confirmed using the platforms in the recent experiments.