In recent years, it had 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-order 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, while the fourth-order NLAH can be the leading response in the surface states of topological insulators induced by Berry curvature hexapole. Our results are established by symmetry analysis, effective Hamiltonian and first-principles calculations. Other materials that support the NLAH effect are further proposed, including 2D antiferromagnets and ferromagnets, Weyl semimetals and twisted bilayer graphene near the quantum anomalous Hall phase.

Recently, a quantum anomalous Hall (QAH) state was observed in AB stacked moiré MoTe2/WSe2 heterobilayers at half-filling. More recent layer-resolved magnetic circular dichroism (MCD) measurements revealed that spin-polarized moiré bands from both the MoTe2 and the WSe2 layers are involved at the formation of the QAH state. This scenario is not expected by existing theories. In this work, we suggest that the observed QAH state is a new state of matter, namely, a topological px + ipy inter-valley coherent state (TIVC). We point out that the massive Dirac spectrum of the MoTe2 moiré bands, together with the Hund's interaction and the Coulomb interactions give rise to this novel QAH state. Through a self-consistent Hartree-Fock analysis, we find a wide range of interaction strengths and displacement fields that the px + ipy-pairing phase is energetically favourable. Besides explaining several key features of the experiments, our theory predicts that the order parameter would involve the pairing of electrons and holes with a definite momentum mismatch such that the pairing would generate a new unit cell which is three times the size of the original moiré unit cell, due to the order parameter modulations.