Effective curved space-time geometric theory of generic-twist-angle graphene with application to a rotating bilayer configuration

We propose a new kind of geometric effective theory based on curved space-time single valley Dirac theory with spin connection for twisted bilayer graphene under generic twist angle. This model can reproduce the nearly flat bands with the particle-hole symmetry around the first magic angle. The bandwidth is near the former results given by Bistritzer-MacDonald model or density matrix renormalization group. Even more, such geometric formalism allows one to predict the properties of rotating bilayer graphene which cannot be accessed by former theories. As an example, we investigate the Bott index of rotating bilayer graphene. We relate this to the two-dimensional Thouless pump with quantized charge pumping during one driving period which could be verified by transport measurements.