Recent theoretical schemes for utilizing cavity QED models as quantum simulators are reviewed. By considering a quadrature representation for the fields, it is shown how Jahn-Teller models, effective Abelian or non-Abelian gauge potentials, transverse Hall currents and relativistic effects naturally arise in these systems. Some of the analytical predictions are verified numerically using realistic experimental parameters taking into account system losses thereby demonstrating their feasibility with current experimental setups.
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