Observation of a ferro-rotational order coupled with second-order nonlinear optical fields

The ferro-rotational order, whose order parameter (OP) is an axial vector invariant under both time reversal (TR) and spatial inversion (SI) operations, is the last remaining category of ferroics to be observed after the ferroelectric, ferromagnetic, and ferro-toroidal orders. This order has become increasingly popular in many new quantum materials, especially in complex oxides, and is considered responsible for a number of novel phenomena such as polar vortices, giant magnetoelectric coupling, and type-II multiferroics. However, physical properties of the ferro-rotational order have been rarely studied either theoretically or experimentally. Here, using high sensitivity rotational anisotropy second harmonic generation (RA SHG), we have, for the first time, exploited the electric quadrupole (EQ) contribution to the SHG to directly couple to this centrosymmetric ferro-rotational order in an archetype of type-II multiferroics, RbFe(MoO4)2. Surprisingly, we have found that two types of domains with opposite ferro-rotational vectors emerge with distinct populations at the critical temperature Tc ~195 K and gradually evolve to reach an even ratio at lower temperatures. Moreover, we have identified the ferro-rotational order phase transition as weak first order, and have revealed its conjugate coupling field as a unique combination of the induced EQ SHG and the incident fundamental electric fields. Our results on physical properties of a ferro-rotational order provide crucial knowledge for understanding and searching for novel phases of matter built upon the ferro-rotational order. Further, these results open the possibility of revealing unconventional centrosymmetric orders and identifying their conjugate coupling fields with second order nonlinear optics.