Electric Ferro-Axial Moment as Nanometric Rotator and Source of Longitudinal Spin Current

An electric ferro-axial moment, which is characterized by a nonzero expectation value of a time-reversal-even axial vector, exhibits distinct spatial-inversion and time-reversal properties from conventional ferroelectric, ferromagnetic, and ferro-magnetoelectric orders. Nevertheless, physical properties characteristic of the electric ferro-axial moment have been obscure owing to the absence of its conjugate electromagnetic fields. We theoretically investigate consequences of the presence of the ferro-axial moment on the basis of the symmetry and microscopic model analyses. We show that atomic-scale electric toroidal multipoles are the heart of the ferro-axial moment, which act as a nanometric rotator against external stimuli. Furthermore, we propose an intrinsic generation of a spin current parallel to an applied electric field in both metals and insulators. Our results not only provide a deep microscopic understanding of the role of the ferro-axial moment but also stimulate a new development for functional materials with use of the electric toroidal moment.