Ferroic orders can be classified by the symmetry of their order parameters, and ferroelectric, ferromagnetic and ferro-toroidal orders have already been observed. The ferro-rotational order^1,2,3, whose order parameter is an axial vector invariant under both time-reversal and spatial-inversion operations, is the final ferroic to be identified and has a vector order parameter. This order is closely related to a number of phenomena such as polar vortices⁴, giant magnetoelectric coupling⁵ and spin-helicity-driven ferroelectricity⁶, but it has received little attention so far. Here, using high-sensitivity rotational-anisotropy second-harmonic generation, we have exploited the electric quadrupole contribution to the second-harmonic generation to directly couple to this centrosymmetric ferro-rotational order in an archetype of type-II multiferroics, RbFe(MoO₄)₂. We found that two domain states with opposite ferro-rotational vectors emerge with distinct populations at the critical temperature T_c ≈ 195 K and gradually evolve to reach an even ratio at lower temperatures. Moreover, we have identified the ferro-rotational order phase transition as weakly first order and have revealed its coupling field as a unique combination of the induced electric quadrupole second-harmonic generation and the incident fundamental electric fields.

铁性有序可以通过其有序参数的对称性进行分类，并且已经观察到铁电，铁磁和铁氧体有序。铁磁旋转阶数^1,2,3（其阶数参数在时间反转和空间反转操作下均为轴向矢量不变）是要识别的最终铁磁并且具有矢量阶数参数。该顺序与许多现象密切相关，例如极涡⁴，巨磁电耦合⁵和自旋螺旋驱动铁电^6。，但到目前为止，它鲜有受到关注。在这里，我们使用高灵敏度旋转各向异性二次谐波生成器，利用对二次谐波生成的电四极贡献，以II型多铁性体RbFe（MoO _4）的原型直接耦合到该中心对称铁旋转序。）₂。我们发现，在临界温度T _c时，两个具有相反铁磁旋转矢量的畴状态出现了不同的种群 ≈195 K，并逐渐演化为在较低温度下达到均匀的比率。此外，我们已将铁旋转阶跃相识别为弱一阶，并揭示了其耦合场是感应电四极二次谐波生成和入射基本电场的独特组合。