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$<em>2$ 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 $\mathbf{\mathcal{O}}<sup>M$ 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 $$\mathbf{P}^{2ω} = \mathrm i\varepsilon</em>{0}{ }^{c\mathbfχ^{{(3)}(\mathbf{\mathcal{O}}^M)}} :\mathbf{E}^ω \nabla \mathbf{E}<sup>ω$$. 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$<sup>{2+}$ 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.