  In recent years, one important experimental achievement was the strong
coupling of quantum matter and quantum light. Realizations reach from ultracold
atomic gases in high-finesse optical resonators to electronic systems coupled
to THz cavities. The dissipative nature of the quantum light field and the
global coupling to the quantum matter leads to many exciting phenomena such as
the occurrence of dissipative quantum phase transition to self-organized exotic
phases. The theoretical treatment of these dissipative hybrid systems of matter
coupled to a cavity field is very challenging. Previously, often mean-field
approaches were applied which characterize the emergence of self-organized
phases as a zero-temperature transition for the particles, a ground-state Dicke
transition. Here we develop a new approach which combines a mean-field approach
with a perturbative treatment of fluctuations beyond mean-field, which becomes
exact in the thermodynamic limit. We argue that these fluctuations are crucial
in order to determine the mixed state (finite temperature) character of the
transition and to unravel universal properties of the self-organized states. We
validate our results by comparing to time-dependent matrix-product-state
calculations.
