Probing magnetic anisotropy in Kagome antiferromagnetic Mn$_3$Ge with torque magnetometry

We investigate the magnetic symmetry of the topological antiferromagnetic material Mn$_3$Ge by using torque measurements. Below the Neel temperature, detailed angle-dependent torque measurements were performed on Mn$_3$Ge single crystals in directions parallel and perpendicular to the Kagome basal plane. The out-of plane torque data exhibit $\pm\sin\theta$ and $\sin2\theta$ behaviors, of which the former results from the spontaneous ferromagnetism within the basal plane and the latter from the in- and out-of-plane susceptibility anisotropy. The reversible component of the in-plane torque exhibits $\sin6\varphi$ behavior, revealing the six-fold symmetry of the in-plane magnetic free energy. Moreover, we find that the free energy minima are pinned to the direction of spontaneous ferromagnetism, which correspond to the maxima of the irreversible component of the in-plane torque. We provide an effective spin model to describe the in-plane magnetic anisotropy. Our results demonstrate that the ground state of Mn$_3$Ge is described by the coexistence of a strong six-fold antichiral order and a weak ferromagnetic order induced by second-order spin anisotropy.