  We discuss the physics of the Rabi-Hubbard model describing large arrays of
coupled cavities interacting with two level atoms via a Rabi non-linearity. We
show that the inclusion of counter-rotating terms in the light-matter
interaction, often neglected in theoretical descriptions based on
Jaynes-Cumming models, is crucial to stabilize finite-density quantum phases of
correlated photons with no need for an artificially engineered chemical
potential. We show that the physical properties of these phases and the quantum
phase transition occurring between them is remarkably different from those of
interacting bosonic massive quantum particles. The competition between photon
delocalization and Rabi non-linearity drives the system across a novel $Z_2$
parity symmetry-breaking quantum phase transition between two gapped phases, a
Rabi insulator and a delocalized super-radiant phase.
