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The Non-linear Hall Acceleration and the Quantum Rectification Sum Rule

We will describe a correction to Newton’s second law when electrons move in Bloch bands without inversion symmetry, whereby, their acceleration acquires a term scaling with the square of the electric field and is orthogonal to it. This non-linear Hall acceleration gives rise to a non-linear Hall effect in time reversal invariant materials that has been recently observed in transition metal dichalcogenides. Such an effect is controlled by the "Berry curvature dipole”, a quantity that plays the role of a non-linear version of the Drude weight in metals without inversion symmetry. We will also discuss a remarkable sum rule for the rectification conductivity of any time reversal invariant band structure which is entirely controlled by the Berry geometry and is independent of the band energies and apply these findings to understand the non-linear optoelectronic properties of Weyl semimetals.