Quantum geometry, Structure factor, and Topological bounds

Topological materials exhibit distinctive physical properties rooted in the topology of the electronic Bloch wave functions. In recent years, the quantum metric, a measure of the distance between quantum states, has attracted growing attention to characterize these wave functions. Together with the Berry curvature, it defines quantum geometry of electronic states, providing a unified geometric framework that complements the topological concepts.

 

In this talk, I will discuss how quantum geometry leads to universal bounds of physical quantities in topological materials — topological bounds. Specifically, I will demonstrate that the static structure factor is directly related to quantum geometry [1] and obeys universal bounds determined by a topological invariant, Chern number [2]. This relation yields a fundamental topological bound on the energy gap of Chern insulators [3]. Remarkably, these results remain exact even in interacting systems, such as fractional Chern insulators. Through these topological bounds, I will illustrate how quantum geometry reveals new and universal aspects of topological phases.

 

[1] Y. Onishi and L. Fu, Quantum weight: A fundamental property of quantum many-body systems, Phys. Rev. Res. 7, 023158 (2025).

[2] Y. Onishi and L. Fu, Topological Bound on the Structure Factor, Phys. Rev. Lett. 133, 206602 (2024).

[3] Y. Onishi and L. Fu, Fundamental Bound on Topological Gap, Phys. Rev. X 14, 011052 (2024).