  In this work, we study the exciton-polariton condensate phase transition in a
microcavity matter-light system in which electron-hole Coulomb interaction and
matter-light coupling effects are treated on an equal footing. In the framework
of the unrestricted Hartree-Fock approximation applying the two-dimensional
exciton-polariton model, we derive the self-consistent equations determining
simultaneously the excitonic and the photonic condenstate order parameters. In
the thermal equilibrium limit, we find a condensed state of the
exciton-polariton systems and phase diagrams are then constructed. At a given
low temperature, the condensate by its nature shows a crossover from an
excitonic to a polaritonic and finally photonic condensed state as the
excitation density increases at large detuning. Without the detuning, the
excitonic condensed state disappears whereas the polaritonic or photonic phases
dominate. The crossover is also found by lowering the Coulomb interaction at a
finite matter-light coupling. Lowering the Coulomb interaction or increasing
the temperature, the excitonic Mott transition occurs, at which the
exciton-polariton condensates dissociate to free electron-hole/photon.
Depending on temperature and excitation density, the phase transition of the
exciton-polariton condensates is also addressed in signatures of
photoluminescence mapping to the photonic momentum distribution.
