  We investigate the potential to induce long-range spin interactions in a Mott
insulator via the quantum electromagnetic field of a cavity. The coupling
between light and spins is inherently non-linear, and occurs via multi-photon
processes like Raman scattering and two-photon absorption/emission with
electronically excited intermediate states. Based on this, two pathways are
elucidated: (i) In the absence of external driving, long-range interactions are
mediated by the exchange of at least two virtual cavity photons. We show that
these vacuum-mediated interactions can surpass local Heisenberg interactions in
mesoscopic setups such as sufficiently small split-ring resonators. (ii) In a
laser-driven cavity, interactions can be tailored through a hybrid scheme
involving both external laser photons and cavity photons. This offers a
versatile pathway for Floquet engineering of long-range interactions in
macroscopic systems. In general, the derivation of these interactions requires
careful consideration: Notably, we demonstrate that a simple phenomenological
approach, based on a spin-photon Hamiltonian that captures Raman and two-photon
processes with effective matrix elements, can be used only if the cavity is
resonantly driven. Outside of these narrow resonant regimes as well as for the
undriven case, a fourth-order series expansion within the underlying electronic
model is necessary, which we perform to obtain long-range four-spin
interactions in the half-filled Hubbard model.
