Dirac cones and magic angles in the Bistritzer--MacDonald TBG Hamiltonian

TL;DR

This paper proves the existence of Dirac cones and magic angles in the full Bistritzer--MacDonald Hamiltonian for twisted bilayer graphene, revealing the stability and topological properties of flat bands and band crossings.

Contribution

It is the first to establish the existence of magic angles for the full Hamiltonian, addressing an open problem and analyzing band crossing behaviors beyond the chiral limit.

Findings

Dirac cones are generically present in the Hamiltonian.

Magic angles correspond to the absence of Dirac cones.

At magic angles, additional band crossings occur beyond quadratic ones.

Abstract

We demonstrate the generic existence of Dirac cones in the full Bistritzer--MacDonald Hamiltonian for twisted bilayer graphene. Its complementary set, when Dirac cones are absent, is the set of magic angles. We show the stability of magic angles obtained in the chiral limit by demonstrating that the perfectly flat bands transform into quadratic band crossings when perturbing away from the chiral limit. Moreover, using the invariance of Euler number, we show that at magic angles there are more band crossings beyond these quadratic band crossings. This is the first result showing the existence of magic angles for the full Bistritzer--MacDonald Hamiltonian and solves Open Problem No.2 proposed in the recent survey arXiv:2310.20642.