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In spite of decades of work it has remained unclear whether or notsuperradiant quantum phases, referred to here as photon condensates, can occurin equilibrium. In this Letter, we first show that when a non-relativisticquantum many-body system is coupled to a cavity field, gauge invariance forbidsphoton condensation. We then present a microscopic theory of the cavity quantumelectrodynamics of an extended Falicov-Kimball model, showing that, inagreement with the general theorem, its insulating ferroelectric and excitoncondensate phases are not altered by the cavity and do not support photoncondensation.

Cavity QED of Strongly Correlated Electron Systems: A No-go Theorem for Photon Condensation