Double and Quadruple Flat Bands tuned by Alternative magnetic Fluxes in Twisted Bilayer Graphene

Twisted bilayer graphene (TBG) can host the moire energy flat bands with two-fold degeneracy serving as a fruitful playground for strong correlations and topological phases. However, the number of degeneracy is not limited to two. Introducing a spatially alternative magnetic field, we report that the induced magnetic phase becomes an additional controllable parameter and leads to the emergence of four-fold degenerate flat bands. This emergence stems from the band inversion at $Gamma$ near the Fermi level as both the twisted angle and the magnetic phase vary. Using holomorphic functions, which explain the origin of the double flat bands in the conventional TBG, we can generate analytical wave functions in the magnetic TBG to show absolutely flat bands with four-fold degeneracy. In contrast, the conventional TBG has only two moire energy flat bands, and the highly degenerate flat bands in this magnetic platform might bring richer correlation physics