Rigid-Body Anisotropy in Noncollinear Antiferromagnets

Characterizing the anisotropic structure in noncollinear antiferromagnets is essential for antiferromagnetic spintronics. In this work, we provide a microscopic theory linking the anisotropy effects induced by the rigid-body rotation of spin order to spin-orbit coupling. Our method goes beyond the conventional magnetic group theory, offering a concise yet powerful tool to characterize diverse anisotropy effects in complex magnetic systems. Using the group representation theory of the spin group, we obtain a set of basis functions formed from tensor elements of spin-orbit vector--which originates from spin-orbit coupling and is tied to the rigid-body rotation of the spin order--to systematically describe the structure of anisotropy effects. As a concrete example, we apply our framework to coplanar antiferromagnets Mn$_3$Sn and Mn$_3$Ir, demonstrating that the corresponding basis functions can well capture both the geometric and magnitude dependencies of the magnetic anisotropy energy and anomalous Hall conductivity. Finally, we discuss the generalization of our framework to broader classes of anisotropy phenomena in magnetic systems.