  Systems of strongly interacting atoms and photons, that can be realized
wiring up individual cavity QED systems into lattices, are perceived as a new
platform for quantum simulation. While sharing important properties with other
systems of interacting quantum particles here we argue that the nature of
light-matter interaction gives rise to unique features with no analogs in
condensed matter or atomic physics setups. By discussing the physics of a
lattice model of delocalized photons coupled locally with two-level systems
through the elementary light-matter interaction described by the Rabi model, we
argue that the inclusion of counter rotating terms, so far neglected, is
crucial to stabilize finite-density quantum phases of correlated photons out of
the vacuum, with no need for an artificially engineered chemical potential. We
show that the competition between photon delocalization and Rabi non-linearity
drives the system across a novel $Z_2$ parity symmetry-breaking quantum
criticality between two gapped phases which shares similarities with the Dicke
transition of quantum optics and the Ising critical point of quantum magnetism.
We discuss the phase diagram as well as the low-energy excitation spectrum and
present analytic estimates for critical quantities.
