Quantum geometry, energy gap and dielectric constant

Abstract

I will present a quantum geometric theory of insulating states that is based on two key quantities: the quantum weight and Chern number defined respectively by the quantum metric and the Berry curvature of the ground state over twisted boundary conditions. As ground state properties, the quantum weight and Chern number are directly measurable through the static structure factor and the quantized dc Hall effect. Remarkably, the quantum weight is also intimately related to the excitation spectrum of the system through the optical sum rules. By combining quantum weight with Chern number and dielectric constant, I derive two universal upper bounds on the energy gap of insulators, and compare them with the measured values.