Polaritonic Bloch's Theorem beyond the Long-Wavelength Approximation

Cavity quantum electrodynamics provides a powerful tool to manipulate material properties, yet it remains a matter of debate whether and how quantized fields affect the periodicity of crystals. Here, we extend Bloch's theorem to crystals under strong light-matter coupling, revealing that polariton quasiparticles preserve lattice periodicity. We introduce a general framework to incorporate multimode cavity fields in a simple and tractable way, showing that additional modes contribute small energy corrections noticeable only at low frequencies. Within the single-photon approximation, these contributions reduce to a spatially uniform effective field in the crystal plane, providing a formal justification for the single-mode and long-wavelength approximations commonly used in molecular polaritonics. Together, these results establish a rigorous framework for describing polaritonic states in crystalline solids.