Universal principles of moiré band structures

Abstract Moiré materials provide a highly tunable environment for the realization of band structures with engineered physical properties. Specifically, moiré structures with Fermi surface flat bands—a synthetic environment for the realization of correlated phases—have moiré unit cells containing thousands of atoms and tantalizingly complex band structures. In this paper we show that statistical principles go a long way in explaining universal physical properties of these systems. Our approach builds on three conceptual elements: the presence of quantum chaos caused by the effective irregularity of the atomic configurations on short length scales, Anderson localization in momentum space, and the presence of approximate crystalline symmetries. Which of these principles dominates depends on material parameters such as the extension of the Fermi surface or the strength of the moiré lattice potential. The phenomenological consequences of this competition are predictions for the characteristic group velocity of moiré bands, a primary indicator for their average flatness. In addition to these generic features, we identify structures outside the statistical context, notably almost flat bands close to the extrema of the unperturbed spectra, and the celebrated zero energy ‘magic angle’ flat bands, where the latter require exceptionally fine tuned material parameters.