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

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-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 ferro-type magnetic octupole 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 P^{2ω}=iϵ_{0}^{c}χ^{(3)}(O^{M}):  E^{ω}∇E^{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^{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.