Quantum-Geometry-Driven Exact Ferromagnetic Ground State in a Nearly Flat Band.

We construct a Hubbard model with a nearly flat band whose quantum geometry can be tuned independent of the energy dispersion and the Coulomb interaction. We show that, when the nearly flat band is half filled, the exact ground state of the model exhibits ferromagnetism and that this ferromagnetism is stabilized by the quantum metric through the spin stiffness. Furthermore, we demonstrate that tuning the quantum geometry alone drives a magnetic phase transition. Our nonperturbative results without resorting to mean-field approximations reveal the quantum-geometric origin of ferromagnetism and the underlying many-body physics in dispersive-band systems.