  The nature of an electric ferro-axial moment characterizing a
time-reversal-even axial-vector quantity is theoretically investigated under
magnetic orderings. We clarify that a vortex spin texture results in the
emergence of the ferro-axial moment depending on the helicity. By introducing a
multipole description, we show that the ferro-axial nature appears when two
types of odd-parity magnetic multipoles become active under the vortex spin
texture: One is the magnetic monopole and the other is the magnetic toroidal
dipole. We present all the magnetic point groups to possess the ferro-axial
moment in the presence of the magnetic orderings with the magnetic monopole and
magnetic toroidal dipole. Moreover, we specifically consider a multi-orbital
model in a four-sublattice tetragonal system to demonstrate the ferro-axial
moment induced by the vortex spin texture. We show that the ferro-axial moment
is microscopically characterized by both the cluster-scale and atomic-scale
axial-vector quantities: The former is described by the vortex of the local
electric dipole and the latter is represented by the outer product of the local
orbital and spin angular momenta. Moreover, we find that the atomic-spin orbit
coupling and the hybridization between orbitals with different parity are key
ingredients to induce the ferro-axial moment. We also apply the result to a
magnetic skyrmion with a topologically-nontrivial spin texture and discuss
candidate materials.
