Hall Conductivity of Chern Insulators in a Static Inhomogeneous Electric Field

We study the Hall conductivity of insulators in a static inhomogeneous electric field to the second order of wave number q by using the Kubo formula. For a Chern insulator with two flat bands, we show that the q² term in hall conductivity is determined by the quantum metric tensor and the Berry curvature. Using a wave-packet semiclassical derivation we provide an intuitive picture to show how the quantum metric tensor comes into play. We also do numerical studies for Chern insulators with nearly flat bands and the Hofstadter model in the small flux limit.