Relations between topology and the quantum metric for Chern insulators

TL;DR

This paper explores the relationship between the topology of two-dimensional Chern insulators and their quantum metric, establishing inequalities and conditions under which quantum volume estimates the Chern number.

Contribution

It introduces the quantum volume of the parameter space and links it to the Chern number, providing new insights into topological characterization of insulators.

Findings

Quantum volume relates to the Chern number via established inequalities.

Inequalities are saturated for fermions filling Landau levels.

Quantum volume can serve as a measure to infer the system's topology.

Abstract

We investigate relations between topology and the quantum metric of two-dimensional Chern insulators. The quantum metric is the Riemannian metric defined on a parameter space induced from quantum states. Similar to the Berry curvature, the quantum metric provides a geometrical structure associated to quantum states. We consider the volume of the parameter space measured with the quantum metric, which we call the quantum volume of the parameter space. We establish an inequality between the quantum volume of the Brillouin zone and that of the twist-angle space. Exploiting this inequality and the inequality between the Chern number and the quantum volume, we investigate how the quantum volume can be used as a good measure to infer the Chern number. The inequalities are found to be saturated for fermions filling Landau levels. Through various concrete models, we elucidate conditions when the quantum volume gives a good estimate of the topology of the system.