Optical phonons as a testing ground for spin group symmetries

Abstract: Lattice vibrations are highly sensitive to crystal symmetries and their changes across phase transitions. The latter can modify irreducible (co)representations and corresponding infrared and Raman selection rules of phonons. This concept is established for relativistic magnetic point groups, simultaneously transforming spatial and spin coordinates. However, in altermagnets described by non-relativistic spin groups with disjunct symmetry operations for both vector spaces, the phonon selection rules have remained unexplored. Here, we present a detailed study of the infrared- and Raman-active modes in the collinear antiferromagnet and altermagnet candidate Co$_2$Mo$_3$O$_8$. Comparing to ab initio calculations accurately capturing the eigenfrequencies, we identify all expected phonon modes at room temperature and deduce their selection rules using both symmetry approaches. Importantly, we observe the change of selection rules upon antiferromagnetic ordering, agreeing with the relativistic symmetry approach, while the spin group formalism predicts no changes. Therefore, optical phonons can reveal the appropriate symmetry treatment.