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Symmetry is closely related to functions in material. When the symmetry is lowered by external stimuli or spontaneous phase transitions, material can become functional. For example, the breaking of spatial inversion (time-reversal) symmetry under ferroelectric (ferromagnetic) ordering leads to an electric (magnetic) property with electric polarization (magnetization). Microscopically, these orderings can be described by the uniform alignment of the electric or magnetic dipole moments.

Another dipole moment, i.e., electric toroidal dipole moment, can exist with neither the breaking of spatial inversion symmetry nor time-reversal symmetry, in contrast to conventional electric and magnetic dipole moments. Electronic ordering characterized by such a dipole moment is referred to as electric ferro-axial (or rotational) ordering. Although ferro-axial ordering is one of the fundamental orderings as well as ordinary electric and magnetic orderings, its nature has been studied less because of the following reasons: Until recently, it lacked a clear quantum mechanical definition and has been discussed in a classical picture as a vortex-type object of electric dipole moments. Further, it does not have conjugate electromagnetic fields owing to spatial inversion and time-reversal symmetries. Thus, ferro-axial ordering was thought to be a boring state without intriguing physical responses.

In this study, we theoretically investigate the consequences of the presence of ferro-axial ordering by incorporating symmetry and microscopic model analyses. Based on the augmented multipole description, we show that atomic-scale electric toroidal multipoles are the heart of ferro-axial ordering, which acts as a nanometric rotator against external stimuli; an input field induces the transverse response of the conjugate physical quantities. Among them, we propose an intrinsic generation of longitudinal spin current by an external electric field, which is a different spin conductive phenomenon from the spin Hall effect. We use a fundamental d-orbital model to clarify the model parameters that are important for inducing this phenomenon. Our results can help in finding unexplored functional material with the electric ferro-axial moment.

(written by Satoru Hayami on behalf of all authors.)