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We introduce the concept of ``quantum geometric nesting'' (QGN) tocharacterize the idealized ordering tendencies of certain flat-band systemsimplicit in the geometric structure of the flat-band subspace. Perfect QGNimplies the existence of an infinite class of local interactions that can beexplicitly constructed and give rise to solvable ground states with variousforms of possible fermion bi-linear order, including flavor ferromagnetism,density waves, and superconductivity. For the ideal Hamiltonians constructed inthis way, we show that certain aspects of the low-energy spectrum can also beexactly computed including, in the superconducting case, the phase stiffness.Examples of perfect QGN include flat bands with certain symmetries (e.g. chiralor time-reversal), and non-symmetry-related cases exemplified with anengineered model for pair-density-wave. Extending this approach, we obtainexact superconducting ground states with nontrivial pairing symmetry.

"Quantum Geometric Nesting'' and Solvable Model Flat-Band Systems