Quantum-Geometric Dipole: A Topological Boost to Flavor Ferromagnetism in Flat Bands

Robust flavor-polarized phases are a striking hallmark of many flat-band moiré materials. In this Letter, we trace the origin of this spontaneous polarization to a lesser-known quantum-geometric quantity: the quantum-geometric dipole. Analogous to how the quantum metric governs the spatial spread of wave packets, we show that the quantum-geometric dipole sets the characteristic size of particle-hole excitations, e.g., magnons in a ferromagnet, which in turn boosts their gap and stiffness. Indeed, the larger the particle-hole separation, the weaker the mutual attraction and the stronger the excitation energy. In topological bands, this energy enhancement admits a lower bound within the local-mode approximation, highlighting the crucial role of topology in flat-band ferromagnetism. We illustrate these effects in microscopic models, emphasizing their generality and relevance to moiré materials. Our results establish the quantum-geometric dipole as a predictive geometric indicator for ferromagnetism in flat bands, a crucial prerequisite for topological order.