Optical phonons as a testing ground for spin group symmetries

TL;DR

This paper investigates how optical phonons can serve as a tool to distinguish between different symmetry frameworks in magnetic materials, especially in altermagnets, by analyzing their infrared and Raman activity.

Contribution

It demonstrates that optical phonon selection rules can differentiate between relativistic and non-relativistic spin group symmetries in magnetic materials.

Findings

Phonon modes match ab initio calculations at room temperature.

Selection rules change with antiferromagnetic ordering, aligning with relativistic symmetry.

Spin group formalism predicts no change in selection rules upon magnetic ordering.

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.