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Low-energy Landau levels of AB-stacked zigzag graphene ribbons

Low-energy Landau levels of AB-stacked zigzag graphene ribbons in the presence of a uniform perpendicular magnetic field (\textbf{B}) are investigated by the Peierls coupling tight-binding model. State energies and associated wave functions are dominated by the \textbf{B}-field strength and the $k_z$-dependent interribbon interactions. The occupied valence bands are asymmetric to the unoccupied conduction bands about the Fermi level. Many doubly degenerate Landau levels and singlet curving magnetobands exist along $k_x$ and $k_z$ directions, respectively. Such features are directly reflected in density of states, which exhibits a lot of asymmetric prominent peaks because of 1D curving bands. The $k_z$-dependent interribbon interactions dramatically modify the magnetobands, such as the lift of double degeneracy, the change of state energies, and the production of two groups of curving magnetobands. They also change the characteristics of the wave functions and cause the redistribution of the charge carrier density. The $k_z$-dependent wave functions are further used to predict the selection rule of the optical transition.