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.