Cavity-induced charge transfer in periodic systems: Length-gauge formalism

We develop a length-gauge formalism for treating one-dimensional periodic lattice systems in the presence of photon cavity inducing light-matter interaction. The purpose is to remove mathematical ambiguities that occur when defining position operator context Power-Zienau-Woolley Hamiltonian. then use diagrammatic approach analyze perturbatively interaction between an electronic quantum system and photonic mode long wavelength. illustrate versatility by studying cavity-induced electric charge imbalance polarization Rice-Mele model with broken inversion symmetry.