Quantum geometry and RKKY in flat bands

Abstract

Flat conduction bands quench the group velocity and thus challenge conventional, dispersion-driven pictures of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, where localized moments are coupled via an effective exchange mediated by conduction electrons. Here we show that RKKY interactions in the flat-band limit are not extinguished by the vanishing group velocity but are instead mediated by the quantum geometry of Bloch states. Starting from a microscopic RKKY derivation, we demonstrate that the Brillouin-zone-averaged quantum metric controls the long-wavelength structure of the static susceptibility, thereby determining the magnetic correlation length and the spin stiffness. As a result, the finite spatial spread of Wannier functions provides an effective long-range coupling channel even when single-particle dispersion is absent. Furthermore, we establish the general principle that the ordering temperature is governed by the quantum metric in finite and low-dimensional samples, effectively circumventing the thermodynamic-limit constraint of the Mermin-Wagner theorem. Specifically, our theoretical investigation reveals that increasing the quantum metric enhances magnetic rigidity and leads to a corresponding rise in the critical temperature within finite-sized systems.