Multipolar Anisotropy in Anomalous Hall Effect from Spin-Group Symmetry Breaking

The traditional view of the anomalous Hall effect (AHE) in ferromagnets is that it arises from the magnetization perpendicular to the measurement plane and that there is a linear dependence on the latter. Underlying such a view is the thinking that the AHE is a time-reversal symmetry breaking phenomenon and can therefore be treated in terms of a power series in the magnetic order. However, this view is squarely challenged by a number of recent experiments, urging for a thorough theoretical investigation on the fundamental level. We find that for strong magnets, it is more appropriate and fruitful to regard the AHE as a spin-group symmetry breaking phenomenon where the critical parameter is the spin-orbit interaction strength, which involves a much smaller energy scale. In collinear ferromagnets, the spin-orbit coupling breaks the ∞2′ spin rotation symmetry, and the key to characterizing such symmetry breaking is the identification of spin-orbit vectors which transform regularly under spin-group operations. Born out of our framework is a rich multipolar relationship between the anomalous Hall conductivity and the magnetization direction, with each pole being expanded progressively in powers of the spin-orbit coupling strength. For the leading order contribution, i.e., the dipole, its isotropic part corresponds to the traditional view, and its anisotropic part can lead to the in-plane AHE where the magnetization lies within the measurement plane. Beyond the dipolar structure, the octupolar structure offers the leading order source of nonlinearity and hence introduces unique anisotropy where the dipolar structure cannot. Our theory thus offers a unified explanation for the in-plane AHE recently observed in various ferromagnets, and further extends the candidate material systems. It can also be generalized to study the anomalous Hall effect in crystals with any periodic spin structure and to study the nonlinear Hall effect and the spin Hall effect. Our theory lays the ground for decoding the coupling between various transport and optical phenomena and the magnetic orders. Published by the American Physical Society 2025