The Hubbard model describes strongly correlated electrons on a lattice. The cavity Hubbard model couples those electrons to a quantized electromagnetic field inside an optical cavity. The cavity photon mediates a new long-range interaction between electrons — mediated by light, not by Coulomb forces — and this interaction competes with the existing electron-electron repulsion.
The phase diagram of the coupled system contains states that neither pure electrons nor pure light would produce. A superradiant antiferromagnetic phase: the electrons develop antiferromagnetic order while simultaneously producing a macroscopic photon condensate. A chiral flux phase: the cavity photon induces circulating currents that break time-reversal symmetry, creating a state where electron motion has a preferred handedness.
The phase transitions between these states exhibit two qualitatively different characters. The superradiant transition from the normal state to the superradiant antiferromagnet is first-order — the photon condensate appears discontinuously. But the transition at the multicritical point where several phases meet is continuous, with Gross-Neveu universality — a universality class associated with relativistic fermion field theories, not with lattice models.
The structural point: the cavity doesn't just modify the electron system — it changes which universality classes are accessible. Without the cavity, the Hubbard model's phase transitions belong to standard condensed-matter universality classes. With the cavity, the photon-mediated interaction introduces a long-range coupling that shifts the phase transitions into universality classes usually associated with high-energy physics. The light doesn't perturb the electrons; it changes the space of phases they can explore.