Supercell-size scaling of moir\'e band flatness

In moir\'e superlattices, the band flatness governs the degree of wave localization, which is central to harnessing emergent phenomena and designing functional meta-devices. While research has focused on the magic conditions such as magic angle and magic distance for optimal flatness, a fundamental understanding of how flatness changes with the supercell size has remained elusive. Here, we establish a universal scaling between band flatness and supercell size. Theoretically, by recognizing the statistical equivalence between structural perturbations in moir\'e superlattices and disordered systems, we introduce the Thouless number to evaluate the strength of moir\'e localization. This approach allows us to establish a scaling theory for the evolution of band flatness with the supercell size, from which an analytical expression is derived. Our full-wave simulations with one-dimensional and two-dimensional moir\'e superlattices show excellent agreement with the theoretical prediction. Our work reveals a general scaling law for moir\'e band flatness, offering a new perspective for understanding and designing moir\'e-based resonant systems.