Multipole moments serve as order parameters for characterizing higher-order magnetic effects in momentum space, providing a framework to describe diverse magnetic responses by extending the concept of magnetism. In this paper, we introduce a methodology to quantitatively determine the multipole moment contributions in anomalous Hall effect through angle-dependent anomalous Hall current, with explicit incorporation of discrete crystal symmetries. Our technique uniquely enables the investigation of octupole contribution in nonperiodic systems, particularly at interfaces and surfaces. Typically, in (${\mathrm{Ag}}_2{\mathrm{Fe}}_5{)}_n$ multilayers with quantum-well-engineered $k$-point selectivity, we observe an octupole-dominated anomalous Hall effect in conventional ferromagnetic materials, through first-principles calculations. These results fundamentally challenge the existing theoretical understanding of the anomalous Hall effect, showing that even the conventional contribution arises not only from the dipole moment (net magnetization). Furthermore, we establish practical control over the octupole contribution through two distinct approaches: interface engineering and magnetic ordering reconfiguration, opening new possibilities for manipulating higher-order transport effects.

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• Anomalous Hall effect
• Multipole order
• Interfaces
• Magnetic multilayers
• First-principles calculations