  In recent years, it has been shown that Berry curvature monopoles and dipoles
play essential roles in the anomalous Hall effect and the nonlinear Hall effect
respectively. In this work, we demonstrate that Berry curvature multipoles (the
higher moments of Berry curvatures at the Fermi energy) can induce higher-order
nonlinear anomalous Hall (NLAH) effect. Specifically, an AC Hall voltage
perpendicular to the current direction emerges, where the frequency is an
integer multiple of the frequency of the applied current. Importantly, by
analyzing the symmetry properties of all the 3D and 2D magnetic point groups,
we note that the quadrupole, hexapole and even higher Berry curvature moments
can cause the leading-order frequency multiplication in certain materials. To
provide concrete examples, we point out that the third-order NLAH voltage can
be the leading-order Hall response in certain antiferromagnets due to Berry
curvature quadrupoles, and the fourth-order NLAH voltage can be the leading
response in the surface states of topological insulators induced by Berry
curvature hexapoles. Our results are established by symmetry analysis,
effective Hamiltonian and first-principles calculations. Other materials which
support the higher-order NLAH effect are further proposed, including 2D
antiferromagnets and ferromagnets, Weyl semimetals and twisted bilayer graphene
near the quantum anomalous Hall phase.
