The weight of the Berry Curvature Dipole

Two independent recent experiments have reported the discovery of the non-linear Hall effect in transition metal dichalcogenides ( arXiv:1809.08744 (2018) and arXiv:1809.09279 (2018)). Within a simple semiclassical model, such non-linear Hall response is controlled by the Berry curvature dipole, which is defined as the average of the gradient of the Berry curvature over the occupied states. In this work we investigate certain deeper microscopic underpinnings of the Berry curvature dipole. We will argue that the Berry dipole plays a role analogous to the Drude weight in the case of non-linear response and quantifies an acceleration of the electron liquid which is second order in electric fields. We will describe new sum rules for the second order electrical response in metals that involve the Berry dipole. We also investigate the role of disorder and show that there are non-linear counterparts to the side-jump corrections with the same scaling with the scattering rate as the Berry dipole contribution which will important when analyzing experiments.