Ferromagnetism versus Antiferromagnetism in Narrow-Band Systems: Competition between Quantum Geometry and Band Dispersion

Magnetism in narrow-band systems arises from the interplay between electronic correlations, quantum geometry, and band dispersion. In particular, both ferro and antiferro magnets are known to occur as ground states of (different) models featuring narrow bands. This poses the question of which is favored and under what conditions. In this Letter, we present a unified theoretical framework to investigate spin physics within narrow bands. By deriving an effective spin model, we show that the nonatomic wave function (quantum geometry) of the narrow bands generally favors ferromagnetic ordering, while band dispersion promotes antiferromagnetic correlations. We find that the competition between these effects gives rise to a tunable magnetic phase and rich spin phenomena. Our approach offers a systematic way to study the magnetic properties of narrow-band systems, integrating the roles of wave function, band structure, and correlation effects.