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.