Quantum-geometric dipole: a topological boost to flavor ferromagnetism in flat bands

TL;DR

This paper introduces the quantum-geometric dipole as a key factor that enhances flavor ferromagnetism in flat bands, linking topology and quantum geometry to magnetic stability in moiré materials.

Contribution

It identifies the quantum-geometric dipole as a novel geometric quantity that predicts ferromagnetic behavior and energy gaps in topological flat-band systems.

Findings

Quantum-geometric dipole influences particle-hole excitation size.

Topology imposes a lower bound on excitation energy.

Results are demonstrated in microscopic models relevant to moiré materials.

Abstract

Robust flavor-polarized phases are a striking hallmark of many flat-band moir\'e materials. In this work, 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 wavepackets, 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 moir\'e materials. Our results establish the quantum-geometric dipole as a predictive geometric indicator for ferromagnetism in flat bands, a crucial prerequisite for topological order.