Generalized Bloch's Theorem for Cavity Exciton Polaron-Polaritons

We show that excitons coupled to cavity photons and phonons admit a generalized Bloch theorem when formulated for the conserved total crystal momentum. In minimal-coupling and Fröhlich representations, the interchange of momenta between fermions and bosons breaks crystalline excitons' translational symmetry. In our symmetry-adapted frame, the Hamiltonian becomes block diagonal, without invoking approximations. The resulting formulation yields dispersions and optical responses of cavity exciton polaron-polaritons, enabling investigations that elucidate material properties in strong coupling.