Determining the Nature of Magnetism in Altermagnetic Candidate RuO$_2$

The terminology "altermagnetism" has recently been adopted to describe collinear magnetic order with no net magnetization and non-relativisitic, momentum-dependent spin-splitting. The archetypal material used to theoretically explore altermagnetism is RuO$_2$, but there has been significant debate as to whether RuO$_2$ possesses magnetic, let alone altermagnetic, order. To address questions surrounding the nature of magnetism in RuO$_2$, we combine symmetry-sensitive torque magnetometry and magnetization measurements in single crystals. The data are inconsistent with collinear magnetic order possessing a N\'eel vector along the $c-$axis. Torque magnetometry further demonstrates an isotropic magnetic susceptibility within the $ab-$plane, indicative of neither a N\'eel vector within the $ab-$plane nor a field-induced N\'eel vector reorientation. Magnetic quantum oscillations from both techniques reveal a nearly spherical Fermi surface pocket at the Brillouin zone center, in agreement with paramagnetic electronic structure calculations. Taken together, these data indicate no detectable long-range magnetic order and, by extension, suggest no altermagnetism in high-quality RuO$_2$ single crystals.