  The anomalous Hall effect (AHE), which provides a bridgeway between the
geometry of quantum wavefunctions 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 originated from
higher-order Berry-curvature multipoles has also been observed in recent
studies. Inspired by the 3rd order AHE and its room temperature sign switching
in kagome antiferromagnet FeSn [S. Sankar, R. Liu, X. Gao, et al, to be
published (2023)], we investigate the generic sign structure of Berry-curvature
induced 3rd 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 3rd order Hall coefficient is dictated
by the interplay between time-reversal symmetry breaking, magnetic order, and
spin-orbit couplings. Our calculations give a possible solution for the "sign
problem" of the 3rd order AHE response in the phase space spanned by the in-
and out-of-plane magnetization, the spin-orbital coupling strength and chemical
potential. We further propose realistic experiment setups to systematically
reveal the sign structure in the 3rd order AHE response via continuously
rotating the magnetic field directions.
