Transverse and orthogonal magnetization in antiferromagnets with higher-rank cluster multipoles

Antiferromagnets under a magnetic field develop the magnetization perpendicular to the field as well as a more conventional one parallel to the field. Up to now, two different sources of perpendicular magnetization (PM) are known. One is the transverse magnetization (TM) induced by the spin canting, and the other is the orthogonal magnetization (OM) arising from the higher-rank cluster multipoles. We firstly point out that TM and OM are indistinguishable by symmetry since PM emerges only when every crystalline symmetry other than order-two antiunitary and inversion symmetries is broken. Nevertheless, we find some reasons that TM and OM can be distinguished. First, OM depends only on cluster multipoles while TM depends also on the dominant spin interaction. We develop a microscopic theory of TM and show that the additional effective field from anisotropic spin interactions gives rise to TM by mixing the transverse to longitudinal spin components. Second, TM is from spin canting and OM is from orbital magnetization. Lastly, we show that both TM and OM contributes to the anomalous planar Hall Effect but distinct angular dependence. Because we explain all previous experimental results by our arguments, we believe that our theory provides a useful guideline for understanding the anomalous magnetic responses of the antiferromagnetic with complex magnetic structures.