The transverse current $\left({\mathbf{j}}_H\right)$ due to the anomalous Hall effect (AHE) is usually assumed to be perpendicular to the magnetization $\left(\mathbf{m}\right)$ in ferromagnetic materials, which governs experiments in spintronics. Generally, this assumption is derived from a continuum model, where the discrete symmetry of the crystal is effectively represented by the concept of an effective mass from the band structure. In this paper, we calculate the spin transport through the nonmagnetic metal $|$ ferromagnetic metal interfaces and find that the corresponding Hall current is generally not perpendicular to $\mathbf{m}$ with only a few exceptions at high-symmetry crystal orientations. The calculation illustrates the breakdown of ${\mathbf{j}}_H=\mathrm{\Theta }\mathbf{m}\times {\mathbf{j}}_c$, where $\mathrm{\Theta }$ denotes the anomalous Hall angle and ${\mathbf{j}}_c$ represents the injecting charge current. An analytical formula based on the discrete symmetry of the solid can describe this effect well. In this framework, the leading order corresponds to the conventional AHE, while higher-order terms account for deviations in the Hall current. Additionally, we identify the presence of a chiral AHE at the interface with odd rotational symmetry (e.g., $C_{3v}$), and the higher-order terms can even dominate the AHE by constructing superlattices. The general existence of hidden chirality in spin transport is also revealed, with a specific focus on interface chirality. Our results highlight the significance of discrete atomic positions in solids for spin transport, which extends beyond the conventional continuum model. Moreover, considering the important application of the AHE in spintronics and the wide existence of the interfaces in the devices, the breakdown of ${\mathbf{j}}_H=\mathrm{\Theta }\mathbf{m}\times {\mathbf{j}}_c$ suggests that all experimental measurements related to the AHE should be re-evaluated, with particular focus on the high-quality van der Waals heterostructures.

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• Anomalous Hall effect
• Chirality
• Transport phenomena
• Interfaces
• First-principles calculations