Lukas Grunwald, Xinle Cheng, Emil Viñas Boström, Michael Ruggenthaler, Marios H. Michael, Dante M. Kennes, Angel Rubio

Interfacing materials with electromagnetic cavities offers a route to modify equilibrium properties through structured vacuum fluctuations. The coupling of light with correlated electrons lacks a characteristic energy scale, making vacuum induced modifications of such systems inherently off-resonant and sensitive to the full photon mode structure. Here, we present a non-perturbative calculation of the cavity induced modification of the magnetic exchange interaction $J$ of the half-filled Hubbard model, including all cavity modes and with parameters determined from first principles. We show that the strength of the modification is controlled by a generalized Purcell factor, proportional to the frequency integrated photonic density of states. This result identifies polaritonic surface cavities as promising platforms to modify correlated systems, while standard Fabry-Pérot resonators produce negligible effects due to spectral weight cancellations upon integration. To perform the calculation, we develop a consistent quantization scheme for materials coupled to a dielectric substrate, in the Coulomb gauge, which reveals a competition between static Coulomb screening and dynamical effects arising from the vector potential. Including both effects is essential to obtain even qualitatively correct predictions. For a gold substrate the light-matter interactions lead to a net enhancement of $J$, whose magnitude is large enough to be observable in two-magnon Raman spectroscopy. Our framework establishes a concrete design principle linking cavity geometry to material response in the off-resonant regime, which will guide future experimental and theoretical explorations.