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

TL;DR

This paper introduces a non-Abelian line graph theory to construct flat bands in multi-orbital systems, enabling the realization of complex flat band phenomena in realistic materials like Kagome metals.

Contribution

It develops a general framework for incorporating higher-angular-momentum orbitals and spin-orbit couplings into line graph models to achieve flat bands.

Findings

Realization of d-orbital flat bands in Kagome lattices.

Bridging the gap between simple lattice models and multi-orbital systems.

A method to incorporate internal degrees of freedom into line graph constructions.

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.