Antiferroquadrupolar Order in Altermagnetic CoF${}_2$

Abstract

Altermagnets host non-relativistic spin-split electronic states whose microscopic origin is theoretically linked to a hidden charge order, yet experimental studies of this charge order are limited. We therefore investigate charge order within CoF${}_2$, a $d$-wave altermagnetic compound with a rutile crystal structure and $\Gamma$-point antiferromagnetism. Combining resonant elastic X-ray scattering, symmetry analysis and ab initio calculations, we directly observe charge ordering and identify it as antiferroquadrupolar in nature. Via electronic structure calculations, we show that the experimentally observed antiferroquadrupolar order gives rise to the characteristic altermagnetic spin-splitting, thereby establishing empirical evidence for the decomposition of the altermagnetic order parameter into magnetic and charge degrees of freedom. We hence demonstrate that antiferroquadrupolar order is the microscopic origin of altermagnetism in CoF${}_2$, with implications to the wider family of rutile altermagnets. Furthermore, our approach is applicable to studying altermagnetism in general, having demonstrated that resonant elastic X-ray scattering can serve as a direct probe of the charge multipoles that underpin spin-split electronic states in these materials.