Kagome Materials II: SG 191: FeGe as a LEGO Building Block for the Entire 1:6:6 series: hidden d-orbital decoupling of flat band sectors, effective models and interaction Hamiltonians*

The electronic structure and interactions of kagome materials such as 1:1 (FeGe class) and 1:6:6 (MgFe6Ge6 class) are complicated and involve many orbitals and bands at the Fermi level. Current theoretical models usually treat the systems in an s-orbital kagome representation, unsuited and incorrect both quantitatively and qualitatively to the material realities. In this work, we lay the basis of a faithful framework of the electronic model for this large class of materials. We show that the complicated ``spaghetti" of electronic bands near the Fermi level can be decomposed into three groups of d-Fe orbitals coupled to specific Ge orbitals. Such decomposition allows for a clear analytical understanding (leading to different results than the simple s-orbital kagome models) of the flat bands in the system based on the S-matrix formalism of generalized bipartite lattices. Our three minimal Hamiltonians can reproduce the quasi-flat bands, van Hove singularities, topology, and Dirac points close to the Fermi level, which we prove by extensive ab initio studies. We also obtain the interacting Hamiltonian of d orbitals in FeGe using the constraint random phase approximation (cRPA) method. We then use this as a fundamental ``LEGO"-like building block for a large family of 1:6:6 kagome materials, which can be obtained by doubling and perturbing the FeGe Hamiltonian. We applied the model to its kagome siblings FeSn and CoSn, and also MgFe6Ge6. Our work serves as the first complete framework for the study of the interacting phase diagram of kagome compounds.