Cavity-controlled symmetry breaking and topological responses in quantum materials

We study quantum materials placed inside chiral or time-reversal-breaking optical cavities, where both photons and electrons must be treated on equal footing. In this setting, the cavity field can mediate effective interactions that inherit the photon's symmetry and imprint it onto the electronic degrees of freedom. We discuss how this mechanism modifies the superconducting order, showing that it can selectively enhance or bias pairing channels that break time-reversal symmetry, thereby acting as a symmetry-filtering channel unavailable in ordinary condensed-matter settings. We further estimate the resulting cavity-mediated energy scale, together with the conditions under which the effect becomes experimentally observable. The cavity-mediated mechanism suggests broader implications for topological phases, and in particular points toward potential control of Hall-type responses in cavity-coupled materials.