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 critical point transition between two distinct with rotational symmetry, band-touching points Berry dipoles opposite charges overlapping in mirror-symmetric way, and carry unique curvature structures leading to special quantization flux. Close such Berry-dipole transitions, we find that extrinsic intrinsic nonlinear Hall conductivity tensors weakly doped regime characterized by universal functions ratio doping level bulk energy gap, directly proportional change invariant across transition. Our work suggests effects display generalized-sense quantized behavior establishing correspondence invariants.