The triadic Hall effect

As an elementary phenomenon, the anomalous Hall effect is indispensable for probing the complexities of fundamental physics, highlighting the spin-orbit coupling and band topology across magnetic materials. The Onsager's reciprocal relation dictates that the anomalous Hall conductivity is an odd function of magnetization, leading to two longstanding impressions that the anomalous Hall conductivity is linear in magnetization and that the magnetization is perpendicular to the Hall-deflection plane. In this talk, we show that the third-order-in-magnetization term can contribute to the anomalous Hall conductivity, which we refer to as the triadic Hall effect. A straightforward and important consequence is that the magnetization can lie within the Hall-deflection plane, leading to the in-plane anomalous Hall effect. Such an effect is purely due to the triadic contribution, as also confirmed by experiments. We further illuminate the role of the spin-orbital coupling in the triadic Hall effect. Our work reveals the intricate relation between the crystal symmetry and band geometry and paves the way of investigating other triadic contributions to various Hall effect.