Geometric bound on structure factor

TL;DR

This paper establishes a geometric bound on the static structure factor in quantum bands, introducing harmonic bands and linking band geometry to topological properties, with implications for fractional Chern insulators.

Contribution

It introduces a geometric bound on the structure factor, characterizes harmonic bands, and connects band geometry with topological bounds relevant for fractional Chern insulators.

Findings

Harmonic bands satisfy a Laplace-like condition.

The geometric bound constrains the $q^4$ term in $S(q)$.

Topological bounds are linked to uniform quantum geometric tensor.

Abstract

We show that a quadratic form of quantum geometric tensor in $k$-space sets a bound on the $q^4$ term in the static structure factor $S(q)$ at small $\vec{q}$. Bands that saturate this bound satisfy a condition similar to Laplace's equation, leading us to refer to them as $\textit{harmonic bands}$. 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 $q^4$ 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.