AI 中文总结
该研究揭示平带系统中量子几何驱动的RKKY相互作用违背单一几何尺度预期，其衰减长度依赖平带几何与带隙，恢复色散后还会出现由几何选择效应导致的特殊RKKY尾。
AI 中文摘要
在平带系统中，量子度量约束着超流体重量、相干长度等物理可观测量，为空间关联提供了单一几何尺度。本文表明，孤立满平带中的RKKY交换会违背这一预期：由于带内通道缺失，交换通过跨越能隙的虚拟带间跃迁进行，其核函数成为实空间平带投影算符与空带投影算符的逆能隙加权迹乘积。因对应动量空间投影算符具有解析性，核函数呈指数衰减，衰减长度ξ_RKKY由复动量平面内解析延拓投影算符的最近奇点决定，故该长度同时依赖平带几何与带隙。将该形式应用于陈平带系统发现，小能隙下ξ_RKKY随量子度量长度非单调变化：先减小后增大，表明更强的量子几何可缩短磁交换范围。恢复色散后，非平凡反演表示可使 gate 调控下反极费米点的布洛赫态重叠消失，产生1/R³的RKKY尾而非常规1/R²的几何选择效应。
英文摘要
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 $ξ_\text{RKKY}$ 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, $ξ_\text{RKKY}$ 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.
Comments5 pages, 4 figures. Comments are welcome
