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The anomalous Hall effect (AHE), which provides a bridgeway between thegeometry of quantum wavefunctions and transport measurements, has been a keyfocus of intensive studies. In addition to the well-studied linear AHE,governed by the electronic Berry curvature, nonlinear AHE originated fromhigher-order Berry-curvature multipoles has also been observed in recentstudies. Inspired by the 3rd order AHE and its room temperature sign switchingin kagome antiferromagnet FeSn [S. Sankar, R. Liu, X. Gao, et al, to bepublished (2023)], we investigate the generic sign structure of Berry-curvatureinduced 3rd order AHE in topological magnetic material. We find that incontrast to the linear Hall coefficient, whose sign is determined by the brokentime-reversal symmetry, the sign of the 3rd order Hall coefficient is dictatedby the interplay between time-reversal symmetry breaking, magnetic order, andspin-orbit couplings. Our calculations give a possible solution for the "signproblem" 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 chemicalpotential. We further propose realistic experiment setups to systematicallyreveal the sign structure in the 3rd order AHE response via continuouslyrotating the magnetic field directions.

The "Sign problem" of the 3rd order anomalous Hall effect in topological magnetic materials