Non-Abelian line graph: A generalized approach to flat bands

Abstract: Flat bands (FBs) in materials can enhance the correlation effects, resulting in exotic phenomena. Line graph (LG) lattices are well known for hosting FBs with isotropic hoppings in $s$-orbital models. Despite their prevalent application in the Kagome metals, there has been a lack of a general approach for incorporating higher-angular-momentum orbitals with spin-orbit couplings (SOCs) into LGs to achieve FBs. Here, we introduce a non-Abelian LG theory to construct FBs in realistic systems, which incorporates internal degrees of freedom and goes beyond $s$-orbital models. We modify the lattice edges and sites in the LG to be associated with arbitrary Hermitian matrices, referred to as the multiple LG. A fundamental aspect involves mapping the multiple LG Hamiltonian to a tight-binding (TB) model that respects the lattice symmetry through appropriate local non-Abelian transformations. We establish the general conditions to determine the local transformations. Based on this mechanism, we demonstrate the realization of $d$-orbital FBs in the Kagome lattice, which could serve as a minimal model for understanding the FBs in transition metal Kagome materials. Our approach bridges the gap between the known FBs in pure lattice models and their realization in multi-orbital systems.