A Generic Topological Criterion for Flat Bands in Two Dimensions
Mutually distorted layers of graphene give rise to a moiré pattern and a variety of non-trivial phenomena. We show that the continuum limit of this class of models is equivalent to a (2+1)-dimensional field theory of Dirac fermions coupled to two classical gauge fields. We further show that the existence of a flat band implies an effective dimensional reduction in the field theory, where the time dimension is ``removed.'' The resulting two-dimensional Euclidean theory contains the chiral anomaly. The associated Atiyah-Singer index theorem provides a self-consistency condition for the existence of flat bands. In particular, it reproduces a series of quantized magic angles known to exist in twisted bilayer graphene in the chiral limit where there is a particle-hole symmetry. We also use this criterion to prove that an external magnetic field splits this series into pairs of magnetic field-dependent magic angles associated with flat moiré-Landau bands. The topological criterion we derive provides a generic practical method for finding flat bands in a variety of material systems including but not limited to moiré bilayers.