  We theoretically investigate electronic orderings with the electric axial
moment without breakings of both spatial inversion and time-reversal symmetries
in the zigzag-chain system. Especially, we elucidate the role of the local
odd-parity hybridization arising from locally noncentrosymmetric lattice
structures based on symmetry and microscopic model analyses. We show that the
odd-parity crystalline electric field gives rise to an effective cross-product
coupling between the electric dipole and electric toroidal dipole, the latter
of which corresponds to the electric axial moment. As a result, the staggered
component of the electric axial moment is induced by applying an external
electric field, while its uniform component is induced via the appearance of
staggered electric dipole ordering. We also show that uniform electric
quadrupole ordering accompanies uniform electric axial moment. Furthermore, we
discuss transverse magnetization as a consequence of the orderings with the
uniform electric axial moment. Our results extend the scope of materials
exhibiting electric axial ordering to those with locally noncentrosymmetric
lattice structures.
