Relations between topology and the quantum metric for Chern insulators

Tomoki Ozawa, Bruno Mera

Published 2021-03-22Version 1

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 Chern number, the quantum volume of the Brillouin zone, and the quantum volume of the twist-angle space. We elucidate conditions under which the inequality becomes saturated. We apply the inequality to various concrete models, including two-dimensional Landau levels, the Haldane model, and multi-band models, and demonstrate that the quantum volume often gives a good estimate of the topology of the system.