Light-Matter Interactions Beyond the Dipole Approximation in Extended Systems Without Multipole Expansion

We present a general theoretical framework to capture light-matter interactions beyond the electric-dipole approximation (EDA), applicable to extended nano- and microscale materials interacting with spatially structured electric fields without truncation at finite multipolar order. The approach is based on the Power-Zienau-Woolley (PZW) Hamiltonian for light-matter interactions and a representation of the material's Hamiltonian in a basis of maximally localized Wannier functions (MLWFs), obtainable from first-principles calculations. We utilize this approach to clarify the limitations of the ubiquitous dipole approximation. We consider electric fields with both uniform and non-uniform intensities and a range of ratios of system size to the wavelength of light. Through this analysis, we identify the conditions under which the EDA breaks down, leading to significant errors in the light-induced dynamics. Contrary to conventional belief, we find that the EDA is remarkably robust for uniformly illuminated 1-D or 2-D materials when light propagates perpendicular to the material. For 3-D materials or non-perpendicular illumination of lower-dimensional materials, conventional wisdom holds and the EDA begins to break down when the wavelength becomes comparable to the system size. Furthermore, the EDA fails when the material is illuminated partially or non-uniformly. For slowly varying field intensities this failure can be corrected by finite-order multipolar corrections. However, for fields that vary substantially, correcting via multipolar terms becomes computationally impractical. In contrast, our approach captures beyond-dipole light-matter interactions at the computational cost of a standard dipole calculation. This efficiency enables accurate first-principles simulations of spatially structured light-matter dynamics in nanoscale devices, quantum materials, and interfaces.