Classification of High-Ordered Topological Nodes Towards MFBs in Twisted Bilayers

At magic twisted angles, Dirac cones in twisted bilayer graphene (TBG) can evolve into flat bands, serving a critical playground for the study of strongly correlated physics. When chiral symmetry is introduced, rigorous mathematical proof confirms that the flat bands are locked at zero energy in the entire Moir\'{e} Brillouin zone (BZ). Yet, TBG is not the sole platform that exhibits this absolute band flatness. Central to this flatness phenomenon are topological nodes and their specific locations in the BZ. In this study, considering TBSs that preserve chiral symmetry, we classify various ordered topological nodes in base layers and all possible node locations across different BZs. Specifically, we constrain the node locations to rotational centers, such as $\Gamma$ and $\text{M}$ points, to ensure the interlayer coupling retains equal strength in all directions. Using this classification as a foundation, we systematically identify the conditions under which MFBs emerge. Additionally, through the extension of holomorphic functions, we provide a proof that flat bands are locked at zero energy, shedding light on the origin of the band flatness. Remarkably, beyond Dirac cones, numerous twisted bilayer nodal platforms can host the flat bands with the degeneracy number more than two, such as four-fold, six-fold, and eight-fold. This multiplicity of degeneracy in flat bands might unveil more complex and enriched correlation physics.