Universal principles of moir\'e band structures

TL;DR

This paper uncovers universal principles governing moiré band structures, emphasizing the roles of quantum chaos, Anderson localization, and symmetries in determining their physical properties, especially band flatness.

Contribution

It introduces a statistical framework explaining the universal features of moiré bands, including the conditions for flat bands and magic angles, based on fundamental physical principles.

Findings

Predicts characteristic group velocities of moiré bands.

Identifies conditions for near-flat and magic angle bands.

Highlights the dominance of different principles depending on material parameters.

Abstract

Moir\'e materials provide a highly tunable environment for the realization of band structures with engineered physical properties. Specifically, moir\'e structures with Fermi surface flat bands - a synthetic environment for the realization of correlated phases - have moir\'e unit cells containing thousands of atoms and tantalizingly complex bands 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 moir\'e lattice potential. The phenomenological consequences of this competition are predictions for the characteristic group velocity of moir\'e 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.