Classification of High-Ordered Topological Nodes towards Moiré Flat Bands in Twisted Bilayers

Twisted bilayer graphene is not the sole platform that exhibits absolute band flatness. Central to this flatness phenomenon are topological nodes and their specific locations in the Brillouin zone. Considering twisted bilayer systems that preserve chiral symmetry, we classify various ordered topological nodes in base layers and all possible node locations across different Brillouin zones. Specifically, we constrain the node locations to rotational centers, such as Γ and 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 Moire flat bands emerge. Additionally, through the extension of holomorphic functions, we provide 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 flat bands with a degeneracy number of 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.