Quantum Geometry and Stability of Moir\'e Flatband Ferromagnetism

TL;DR

This paper investigates the stability of moiré flatband ferromagnetism by analyzing collective excitations and quantum geometry effects, deriving an analytical expression for spin stiffness, and exploring skyrmion excitations in relation to band topology.

Contribution

It provides a new analytical expression linking spin stiffness to quantum geometric quantities and reveals the role of Berry curvature in magnon stability and skyrmion-electron binding.

Findings

Berry curvature enhances spin magnon stiffness.

Analytical formula for spin stiffness involving quantum geometry.

Skyrmions bind electrons proportional to Chern number and winding number.

Abstract

Several moir\'e systems created by various twisted bilayers have manifested magnetism under flatband conditions leading to enhanced interaction effects. We theoretically study stability of moir\'e flatband ferromagnetism against collective excitations, with a focus on the effects of Bloch band quantum geometry. The spin magnon spectrum is calculated using different approaches, including Bethe-Salpeter equation, single mode approximation, and an analytical theory. One of our main results is an analytical expression for the spin stiffness in terms of the Coulomb interaction potential, the Berry curvatures, and the quantum metric tensor, where the last two quantities characterize the quantum geometry of moir\'e bands. This analytical theory shows that Berry curvatures play an important role in stiffening the spin magnons. Furthermore, we construct an effective field theory for the magnetization fluctuations, and show explicitly that skyrmion excitations bind an integer number of electrons that is proportional to the Bloch band Chern number and the skyrmion winding number.