Exactly Solvable Magnetic Phase Transition and Spin Stiffness in (Generalized) Kagome lattices

The kagome lattice Hubbard model exhibits a strongly interacting gapless flat band with divergent quantum geometry and is an ideal platform for the theoretical and experimental study of correlated topological materials. We show that a large family of similar Hubbard models have exactly solvable ground states up to half filling of the flat bands, generalizing a result of Mielke and Tasaki. We prove a lower bound on the critical density at which these models exhibit a magnetic phase transition, with possible relevance to the onset of ferromagnetism observed in twisted MoTe₂. Lastly, we reveal the importance of band geometry to the spin wave stiffness at half filling, where interactions regularize a naively divergent quantum geometric contribution. We explain a duality relating this result to the Cooper pair mass in flat band superconductors.