Relations between topology and the quantum metric for Chern insulators

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.

Automated summary

This study investigates the relationship between topology and quantum metric in two-dimensional Chern insulators, proposing the concept of quantum volume as a measure for inferring the Chern number. Through inequalities and concrete models, it is concluded that quantum volume can provide a good estimate of the system's topology, especially for fermions filling Landau levels.