Quantum Geometry-Driven RKKY: From Flat to Dispersive Bands
In flat-band systems, quantum metric bounds physical observables like superfluid weight and coherence length, suggesting a single geometric scale for spatial correlations. Here, we show that the RKKY exchange in an isolated filled flat band can violate this expectation. With the intraband channel absent, the exchange proceeds via virtual interband transitions across the gap; the kernel becomes the inverse-gap-weighted trace product of the real-space flat-band projector and empty-band projectors. Because the corresponding momentum-space projectors are analytic, the kernel decays exponentially, with a decay length $\xi_\text{RKKY}$ set by the closest singularities of the analytically continued projectors in the complex momentum plane. Thus, this length depends on both the flat-band geometry and the band gap. Applying this formalism to Chern flat-band systems, we find that for small gaps, $\xi_\text{RKKY}$ depends non-monotonically on the quantum metric length: it first decreases, then increases, revealing that stronger quantum geometry can shorten the magnetic exchange range. Upon restoring dispersion, a nontrivial inversion representation can force the overlap between Bloch states at antipodal Fermi points to vanish under gate tuning, producing a $1/R^3$ RKKY tail instead of the conventional $1/R^2$---a geometric selection effect.