Magic angles in twisted bilayer graphene near commensuration: towards a hyper-magic manifold

Michael G. Scheer, Kaiyuan Gu, Biao Lian

Published 2022-03-11Version 1

The Bistritzer-MacDonald continuum model (BM model) describes the low-energy moir\'e bands for twisted bilayer graphene (TBG) at small twist angles. We derive a generalized continuum model for TBG near any commensurate twist angle, which is characterized by a complex inter-layer hopping at commensurate $AA$ stackings (rather than the real hopping in the BM model), a real inter-layer hopping at commensurate $AB/BA$ stackings, and a global energy shift. The complex phase of the $AA$ stacking hopping and the twist angle together define a single angle parameter $\phi_0$. We compute the model parameters for the first six distinct commensurate TBG configurations, among which the $38.2^\circ$ configuration may be within experimentally observable energy scales. We identify the first magic angle for any $\phi_0$ at a condition similar to that of the BM model. At this angle, the lowest two moir\'e bands at charge neutrality become flat (except in the vicinity of the $\boldsymbol\Gamma_M$ point) and retain fragile topology, but lose particle-hole symmetry. We further identify a hyper-magic manifold in the parameter space at $\phi_0=\pm\pi/2$, where seven or more moir\'e bands around charge neutrality become flat simultaneously. The lowest two moir\'e flat bands in the hyper-magic manifold have fragile (trivial) topology when close to (far from) the chiral limit with zero $AA$ hopping.