I will first recall the Rabi and the related Jaynes-Cummings models after which I will present the Rabi lattice (RLM) and the Jaynes-Cummings (JC) lattice models. After a brief review of the JC model phase diagram, I will present our work on the phase diagram of the RLM using mean field and QMC simulations. We will map out the  phase diagram and study the role of the counter rotating terms and the conditions under which the RLM approaches the JC model.

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