Third-order anomalous Hall response to the Fermi-surface topology in magnetic materials

The anomalous Hall effect (AHE), which provides a bridge between the geometry of quantum wave functions and transport measurements, has been a key focus of intensive studies. In addition to the well-studied linear AHE, governed by the electronic Berry curvature, nonlinear AHE originating from higher-order Berry-curvature multipoles has also been observed in recent studies. Inspired by the third-order AHE and its room temperature sign switching in kagome antiferromagnet FeSn, we investigate the generic sign structure of Berry-curvature-induced third-order AHE in topological magnetic material. We find that in contrast to the linear Hall coefficient, whose sign is determined by the broken time-reversal symmetry, the sign of the third-order Hall coefficient is dictated by the interplay between time-reversal symmetry breaking, magnetic order, and spin-orbit couplings. Our calculations give a picture using third-order AHE response to identify the Fermi-surface topology in the phase space spanned by the in- and out-of-plane magnetization, the spin-orbital coupling strength, and the chemical potential. We further propose realistic experiment setups to systematically reveal the sign structure in the third-order AHE response via continuously rotating the magnetic field directions.