  Electrons moving in a Bloch band are known to acquire an anomalous Hall
velocity proportional to the Berry curvature of the band which is responsible
for the intrinsic linear Hall effect in materials with broken time-reversal
symmetry. Here, we demonstrate that there is also an anomalous correction to
the electron acceleration which is proportional to the Berry curvature dipole
and is responsible for the Nonlinear Hall effect recently discovered in
materials with broken inversion symmetry. This allows us to uncover a deeper
meaning of the Berry curvature dipole as a nonlinear version of the Drude
weight that serves as a measurable order parameter for broken inversion
symmetry in metals. We also derive a quantum rectification sum rule in time
reversal invariant materials by showing that the integral over frequency of the
rectification conductivity depends solely on the Berry connection and not on
the band energies. The intraband spectral weight of this sum rule is exhausted
by the Berry curvature dipole Drude-like peak, and the interband weight is also
entirely controlled by the Berry connection. This sum rule opens a door to
search for alternative photovoltaic technologies based on the Berry geometry of
bands. We also describe the rectification properties of Weyl semimetals which
are a promising platform to investigate these effects.
