Abstract
We show that a quadratic form of quantum geometric tensor in k-space sets a bound on the q⁴ term in the static structure factor S(q) at small q. Bands that saturate this bound satisfy a condition similar to Laplace's equation, leading us to refer to them as harmonic bands. We provide examples of harmonic bands in one- and two-dimensional systems, including (higher) Landau levels. The geometric bound further leads to a topological bound on the q⁴ term, which is saturated only when the band geometry satisfies the trace condition and, additionally, the quantum geometric tensor is uniform in k-space. We speculate that these bounds taken together provide a useful guide for identifying Chern bands that favor (Abelian or non-Abelian) fractional Chern insulators.