Topology and Spectral Entanglement in Cavity-Mediated Photon Scattering

Abstract: We develop a diagrammatic theory of cavity-mediated photon-photon interactions in a topological insulator using the SSH model. The fourth-order vertex $\Gamma^{(4)}(\omega_1,\omega_2)$ governs spectral entanglement and Kerr nonlinearity, leading to a nonlinear topological phase diagram. We also compute the electronic self-energy from vacuum photon exchange and identify symmetry-imposed limits on band renormalization. These results link band geometry to the emergence of light-matter correlations.