  Can a generic magnetic insulator exhibit a Hall current? The quantum
anomalous Hall effect (QAHE) is one example of an insulating bulk carrying a
quantized Hall conductivity and other insulators (with zero Chern number)
present zero Hall conductance in the linear response regime. Here, we find that
a general magnetic insulator possesses a nonlinear Hall conductivity quadratic
to the electric field if the system breaks inversion symmetry. This
conductivity originates from an induced orbital magnetization due to virtual
interband transitions. We identify three contributions to the wavepacket
motion, a velocity shift, a positional shift, and a Berry curvature
renormalization. In contrast to the crystalline solid, we find that this
nonlinear Hall conductivity vanishes for Landau levels of a 2D electron gas,
indicating a fundamental difference between the QAHE and the Integer quantum
Hall effect.
