Graph Theorem for Chiral Exact Flat Bands at Charge Neutrality

Chiral exact flat bands (FBs) at charge neutrality have attracted much recent interest, presenting an intriguing condensed-matter system to realize exact many-body phenomena, as specifically shown in "magic angle" twisted bilayer graphene for superconductivity and triangulene-based superatomic graphene for excitonic condensation. Yet, no generic physical model to realize such FBs has been developed. Here we present a new mathematical theorem, called bipartite double cover (BDC) theorem, and prove that the BDC of line-graph (LG) lattices hosts at least two chiral exact FBs of opposite chirality, i.e., yin-yang FBs, centered-around/at charge neutrality (E = 0) akin to the "chiral limit" of twisted bilayer graphene. We illustrate this theorem by mapping it exactly onto tight-binding lattice models of the BDC of LGs of hexagonal lattice for strong topological and of triangular lattice for fragile topological FBs, respectively. Moreover, we use orbital design principle to realize such exotic yin-yang FBs in non-BDC lattices to instigate their real material discovery. This work not only enables the search for exact chiral FBs at zero energy beyond moir\'e heterostructures, but also opens the door to discovering quantum semiconductor features with FB-enabled strongly correlated carriers.