Role of the Fermi ring in determining the magic angles in twisted bilayer graphene

The unexpected discovery of superconductivity and strong electron correlation in twisted bilayer graphene (TBG), a system containing only $sp$ electrons, is considered as one of the most intriguing developments in two-dimensional materials in recent years. The key feature is the emergent flat energy bands near the Fermi level, a favorable condition for novel many-body phases, at the so-called ``magic angles''. The physical origin of these interesting flat bands has been elusive to date, hindering the construction of an effective theory for the unconventional electron correlation. In this paper, we have identified the importance of charge accumulation in the AA region of the moir\'e supercell and the most critical role of the Fermi ring in AA-stacked bilayer graphene. We show that the magic angles can be predicted by the moir\'e periodicity determined by the size of this Fermi ring. The resonant criterion in momentum space makes it possible to coherently combine states on the Fermi ring through scattering by the moir\'e potential, leading to flat bands near the Fermi level. We thus establish the physical origin of the magic angles in TBG and identify the characteristics of one-particle states associated with the flat bands for further many-body investigations.