Topology and Spectral Entanglement in Cavity-Mediated Photon Scattering

We develop a diagrammatic theory of cavity-mediated photon-photon interactions in a topological insulator using the SSH model. The fourth-order vertex ${\mathrm{\Gamma }}^{(4)}({\omega }_1,{\omega }_2)$ 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.