Magic angles in twisted bilayer graphene near commensuration: Towards a hypermagic regime

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 complex interlayer hoppings at commensurate $AA$ stackings (rather than the real hoppings in the BM model), a real interlayer hopping at commensurate $AB/BA$ stackings, and a global energy shift. The complex phases of the $AA$ stacking hoppings 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 near the $\boldsymbol\Gamma_M$ point and retain fragile topology but lose particle-hole symmetry. We further identify a hypermagic parameter regime centered at $\phi_0 = \pm\pi/2$ where many moir\'e bands around charge neutrality (often $8$ or more) become flat simultaneously. Many of these flat bands resemble those in the kagome lattice and $p_x$, $p_y$ 2-orbital honeycomb lattice tight-binding models.