Fig. 1. (a) Schematic illustration of the moiré pattern in twisted bilayer graphene, the inserted shows the wrinkles of the graphene for different layer distances; (b) schematic illustration of twisted double bilayer graphene system; (c) moiré Brillouin zone of twisted graphene systems.

Fig. 2. (a) Band structures of twisted bilayer graphene at the magic angle; (b)Wilson loops of the two flat bands from the K valley of twisted bilayer graphene (in ).

Fig. 3. (a) Band structures of twisted double bilayer graphene at ; (b) the Chern number of the first conduction band for the K valley of AB-AB stacked twist-ed double bilayer graphene vs. the twist angle and vertical potential drop .

Fig. 4. Real-space current-density distribution contributed by the flat bands of magic-angle twisted bilayer graphene: (a) K valley, ; (b) valley, ; (c) K valley, ; (d) valley, . The black arrows indicate the dire-ctions of the current density, and the color coding indicates the magnetic field induced by the current in unites of Tesla.

Fig. 5. (a) The orbital magnetization contributed by the flat bands of the K valley of twisted bilayer graphene at the magic angle; (b) the orbital magnetization contributed by the flat bands of K valley for twisted double bilayer graphene at .