Exactly flat bands beyond the chiral limit in Bistritzer--MacDonald Hamiltonians

Abstract

The chiral limit of the Bistritzer--MacDonald model for twisted bilayer graphene has exactly flat bands at magic angles. We show that exact flatness can persist beyond the chiral limit with nonchiral tunnelling potentials in the usual symmetry class. We construct a family of nonchiral tunnelling potentials for which the Hamiltonian has at least two zero-energy states at every Bloch momentum and for every real value of the nonchiral coupling. This gives a family of counterexamples to Open Problem~3 of Zworski's survey \cite{ZworskiSurvey}. Conversely, we prove that every admissible potential with this property belongs to this family, thereby obtaining a complete characterization. By contrast, we show that the standard BM model, namely, BM Hamiltonians with first harmonic tunnelling potentials, does not admit exactly flat bands at the magic angle except possibly at a discrete set of nonchiral coupling strengths, with no finite accumulation point. To prove this, we introduce an explicit invertibility criterion. The criterion requires the projected perturbation to have a nonzero scalar coefficient at some Bloch momentum, obstructing exact flatness for general nonchiral tunnelling. A local Taylor expansion argument verifies the criterion for the standard BM potentials. For the family constructed above, however, the projected perturbation vanishes at every Bloch momentum, so the invertibility criterion fails.