  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.
