We explore theoretically the nonequilibrium photonic phases of an array of coupled cavities in presence of incoherent driving and dissipation . In particular, we consider a Hubbard model system where each site is a Kerr nonlinear resonator coupled to a two-level emitter, which is pumped incoherently . Within a Gutzwiller mean-field approach, we determine the steady-state phase diagram of such a system.
We find that, at a critical value of the inter-cavity photon hopping rate, a second-order nonequilibrium phase transition associated with the spontaneous breaking of the $U(1)$ symmetry occurs.
The transition from an incompressible Mott-like photon fluid to a coherent delocalized phase is driven by commensurability effects and not by the competition between photon hopping and optical nonlinearity.
The essence of the mean-field predictions is corroborated by finite-size simulations obtained with matrix product operators and corner-space renormalization methods.
 A. Biella, F. Storme, J. Lebreuilly, D. Rossini, R. Fazio, I. Carusotto, and C. Ciuti, Phase diagram of incoherently-driven strongly correlated photonic lattices, arXiv:1704.08978 (2017).
 J. Lebreuilly, M. Wouters, and I Carusotto, Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pumping, Comptes Rendus Physique 17, 836 (2016).