Flat-band systems feature rich interplay between quantum geometry/topology encoded in the single-particle wavefunctions and strong interactions relative to the small bandwidth. The conventional paradigms suitable for studying weakly correlated systems can not be well justified here. Instead, strong coupling theories are desirable, for which the identification of solvable limits is crucial.

In [1], we provide a systematic criterion to determine whether a given quantum geometry (i.e. the single particle wavefunctions) is suitable for any form of fermion bilinear orders, and if so, what are the ideal interactions that can lead to them. These ideal models have fully solvable ground states and few-body excitations, rendering convenient handles for the study of realistic models in the closeby parameter regime. 

Recently, we further generalized this concept to pairing in flavorless flat bands [2], which cannot be treated within the original framework. A key result is surprisingly simple: in flat bands with a generalized nesting condition ensured by inversion symmetry, any local attraction between orbitals of opposite parity is exactly solvable, yielding a superconducting ground state whose topology is directly dictated by the parent flat band. Since this attraction occurs between distinct orbitals rather than onsite, it offers a realistic route toward topological superconductivity, as strong Coulomb repulsion typically causes flavor polarization while leaving residual attractive channels. These models host an exact particle–hole duality within the flat-band subspace, producing double superconducting domes as a function of electron density and charge-transfer gap—remarkably reminiscent of experimental observations in rhombohedral multilayer graphene with 4–6 layers.