Non-uniform electric response in Chern insulators

We consider the current response of finite width strips in Chern insulator systems with non-flat Berry curvature. The response to a slowly spatially varying electric potential of the hybrid Wannier centers in the strip gives the charge polarization to lowest sub-leading order in the external field. We show how this is a result of the quantum geometry of the Chern insulator. Higher order response may provide insight into the projected density algebra of the Chern insulator which controls fractional state stability.