Second-order topological insulators (SOTIs) are a class of materials hosting gapless bound states at boundaries with dimension lower than the bulk by two. In this work, we investigate the effect of Zeeman field on two- and three-dimensional time-reversal invariant SOTIs. We find that a diversity of topological phase transitions can be driven by the Zeeman field, including both boundary and bulk types. For boundary topological phase transitions, we find that the Zeeman field can change the time-reversal invariant SOTIs to time-reversal symmetry breaking SOTIs, accompanying with the change of the number of robust corner or hinge states. Relying on the direction of Zeeman field, the number of bound states per corner or chiral states per hinge can be either one or two in the resulting time-reversal symmetry breaking SOTIs. Remarkably, for bulk topological phase transitions, we find that the transitions can result in Chern insulator phases with chiral edge states and topological semimetal phases with sharply-localized corner states in two dimensions, and hybrid-order Weyl semimetal phases with the coexistence of surface Fermi arcs and gapless hinge states in three dimensions. Our study reveals that the Zeeman field can induce very rich physics in higher-order topological materials.

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.

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