  Effects of spontaneous toroidal ordering on magnetic excitation are
theoretically investigated for a localized spin model that includes a staggered
Dzyaloshinsky-Moriya interaction and anisotropic exchange interactions, which
arise from the antisymmetric spin-orbit coupling and the multi-orbital
correlation effect. We show that the model exhibits a Neel-type
antiferromagnetic order, which simultaneously accompanies a ferroic toroidal
order. We find that the occurrence of toroidal order modulates the magnon
dispersion in an asymmetric way with respect to the wave number: a toroidal
dipole order on the zigzag chain leads to a band-bottom shift, while a toroidal
octupole order on the honeycomb lattice gives rise to a valley splitting. These
asymmetric magnon excitations could be a source of unusual magnetic responses,
such as nonreciprocal magnon transport. A variety of modulations are discussed
while changing the lattice and magnetic symmetries. Implications of candidate
materials for asymmetric magnon excitations are presented.
