  Three-dimensional Hopf insulators are a class of topological phases beyond
the tenfold-way classification. The critical point separating two
rotation-invariant Hopf insulator phases with distinct Hopf invariants is quite
different from the usual Dirac-type or Weyl-type critical points and uniquely
characterized by a quantized Berry dipole. Close to such Berry-dipole
transitions, we find that the extrinsic and intrinsic nonlinear Hall
conductivity tensors in the weakly doped regime are characterized by two
universal functions of the ratio between doping level and bulk energy gap, and
are directly proportional to the change in Hopf invariant across the
transition. Our work suggests that the nonlinear Hall effects display a
general-sense quantized behavior across Berry-dipole transitions, establishing
a correspondence between nonlinear Hall effects and Hopf invariant.
