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Nonlinear Optical Probing of Ferroic Octupolar Order Parameter in Collinear Altermagnet

P. A. Usachev , R. V. Pisarev , and V. V. Pavlov

Ioffe Institute, Saint Petersburg, 194021, Russia

Phys. Rev. Lett. 137, 126902 – Published 14 September, 2026

DOI: https://doi.org/10.1103/h1ht-vsrn

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-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 $P^{2 ω} = i ϵ_{0}^{c} χ^{(3)} (O^{M}) : E^{ω} \nabla E^{ω}$ . 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.

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Nonlinear Optical Probing of Ferroic Octupolar Order Parameter in Collinear Altermagnet

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