Effective theory for matter in non-perturbative cavity QED

Starting from a general material system of $N$ particles coupled to cavity, we use coherent-state path integral formulation produce non-perturbative effective theory for the degrees freedom. We tackle effects image charges, $A^2$ term and multimode arbitrary-geometry cavity. The resulting (non-local) action has photonic freedom replaced by an position-dependent interaction between particles. In large-$N$ limit, discuss how can be cast into Hamiltonian where cavity induced interactions are made explicit. is applicable, beyond QED, any bulk linearly diagonalizable bosonic bath. highlight differences with other well-known methods numerically study its finite-size scaling on Dicke model. Finally, showcase descriptive power three examples: photon condensation, 2D free electron gas in modification magnetic molecular spins; recovering, condensing extending some recent results literature.