FeGe as a building block for the kagome 1:1, 1:6:6, and 1:3:5 families: hidden d-orbital decoupling of flat band sectors, effective models and interaction Hamiltonians

The electronic structure and interactions of kagome materials, such as the 1:1 (FeGe) and 1:6:6 (MgFe6Ge6) classes, are complicated and involve many orbitals and bands around the Fermi level. Current theoretical models 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 Fe d orbitals coupled to specific Ge orbitals via symmetry and chemical analysis. Such a 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 quasiflat 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 for the d orbitals in FeGe using the constraint random phase approximation (cRPA) method, which faithfully describes the antiferromagnetic phase. We then use FeGe 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 apply the model to its kagome siblings FeSn and CoSn, and also MgFe6Ge6. We further extend the formalism developed for the 1:1 family to the 1:3:5 family AB3Z5 (A = K, Rb, Cs; B = Cr, V, Ti; Z = Sb, Bi), demonstrating the broad applicability of the LEGO-like building block approach. Our work serves as the first complete framework for the study of the interacting phase diagram of kagome compounds.