Electric toroidal octupolar symmetry in pyrite FeS${}_2$ probed by Raman optical activity

Abstract

We report Raman optical activity in pyrite FeS${}_2$, which hosts an electric toroidal octupolar symmetry. A clear and reproducible sign reversal of the circular intensity difference is observed between neighboring $\{111\}$ faces under cross-circular polarization. The signal appears only for the doubly degenerate $E_g$ phonon mode and is absent for other modes, consistent with symmetry analysis. First-principles calculations reproduce these features, establishing Raman optical activity as a probe of higher-rank axial multipolar symmetry.