Generalized Peierls substitution for the tight-binding model of twisted graphene systems in a magnetic field

This work aims at addressing an important advanced methodology for twisted graphene in the presence of applied magnetic field, which is the Bloch-basis tight-binding model (TBM) in conjunction with the generalized Peierls substitution. We investigate extensively the band structures, Landau levels (LLs), and quantum Hall conductivity (QHC) of twisted bilayer graphene and twisted double-bilayer graphene, as well as their dependence on the twist angle. Comparison between these crucial properties of monolayer graphene, Bernal bilayer graphene, and the twisted systems is carefully made to highlight the roles played by twisting. The unique selection rules of inter-LL transition, which is crucial for achieving a deep understanding of the step structures of QHC, are identified through the properties of LL wave functions. The effective TBM is combined with the generalized Peierls substitution to investigate the magnetic quantization of twisted graphene systems at magic angle. Our theoretical model opens up an opportunity for comprehension of the interplay between an applied magnetic field and the twisting effect associated with layered graphene. The proposed method is expected to be applicable for the calculation of magnetic quantization problems of other complex systems.