Microscopic theory of phonon polaritons and dielectric response in the long-wavelength limit

We present a first-principles approach to calculate phonon-polariton dispersion relations. Starting from a Hamiltonian that describes harmonic lattice vibrations coupled with the electromagnetic field, we quantize the system and derive a Hamiltonian for the coupled phonon-photon system. Through a unitary transformation, we diagonalize the Hamiltonian to obtain non-interacting polariton quasiparticles.

Our method naturally includes retardation effects, resolving non-analytical behavior and ambiguities in phonon frequencies at the Brillouin zone center, especially in non-cubic and optically anisotropic materials. By incorporating higher-order terms in the Hamiltonian, we account for quasiparticle interactions and spectral broadening with parameters obtained from first-principles calculations.

We demonstrate how anharmonic effects in phonon polaritons lead to a dielectric response that challenges traditional models.