Extrinsic and intrinsic nonlinear Hall effects across Berry-dipole transitions

Three-dimensional Hopf insulators are a class of topological phases beyond the tenfold-way classification. At the critical point of the transition between two distinct Hopf insulators with rotational symmetry, the band-touching points are point Berry dipoles with two opposite Berry charges overlapping in a mirror-symmetric way, and carry unique Berry curvature structures leading to a special quantization of Berry flux. 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 generalized-sense quantized behavior across Berry-dipole transitions, establishing a correspondence between nonlinear Hall effects and Hopf invariants.