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Systems of strongly interacting atoms and photons, that can be realizedwiring up individual cavity QED systems into lattices, are perceived as a newplatform for quantum simulation. While sharing important properties with othersystems of interacting quantum particles here we argue that the nature oflight-matter interaction gives rise to unique features with no analogs incondensed matter or atomic physics setups. By discussing the physics of alattice model of delocalized photons coupled locally with two-level systemsthrough the elementary light-matter interaction described by the Rabi model, weargue that the inclusion of counter rotating terms, so far neglected, iscrucial to stabilize finite-density quantum phases of correlated photons out ofthe vacuum, with no need for an artificially engineered chemical potential. Weshow that the competition between photon delocalization and Rabi non-linearitydrives the system across a novel $Z_2$ parity symmetry-breaking quantumcriticality between two gapped phases which shares similarities with the Dicketransition of quantum optics and the Ising critical point of quantum magnetism.We discuss the phase diagram as well as the low-energy excitation spectrum andpresent analytic estimates for critical quantities.

On the phase transition of light in cavity QED lattices