Nonlinear Optical Probing of Ferroic-Octupolar Order Parameter in Collinear Altermagnet

Abstract

Altermagnetism as a new concept in condensed matter physics is currently being thoroughly investigated. Despite the absence of macroscopic magnetization, altermagnets host a hidden spin order whose direct detection remains a challenge. Here we report on the observation of electric and magnetic dipole forbidden optical second-harmonic generation (SHG) in the altermagnet CoF${}_2$ with a centrosymmetric lattice and spin order. We demonstrate that below the Néel temperature $T_N=38$ K the SHG signal is sensitive to the ferrotype magnetic octupole ${\mathcal{O}}^M$ which is the order parameter in the antiferromagnetic phase. By combining polarization-resolved SHG experimental data and a phenomenological symmetry analysis, we show that the altermagnetic spin structure of CoF${}_2$ enables a ferroic-octupole-induced electric-quadrupolar nonlinear polarization ${\mathbf{P}}^{2\omega }=\mathrm{i}{\epsilon }_0{}^c{\chi }^{(3)}({\mathcal{O}}^M):{\mathbf{E}}^{\omega }\nabla {\mathbf{E}}^{\omega }$. The temperature dependence of SHG reveals a phase transition at $T_N$ confirming the spin origin of the observed signal. The SHG response is resonantly enhanced by a coherent three-photon process caused by the electronic $d$-$d$ transitions of the Co${}^{2+}$ ion. Model calculations of SHG polarization rotational anisotropies and temperature dependencies give a reasonable agreement with experimental data, proving the disclosed nonlinear contribution. Our results establish SHG as a novel sensitive tool of ferroic-octupolar spin ordering and highlight the potential of CoF${}_2$ and other altermagnets for further nonlinear optical investigations and applications.