  We investigate a microscopic origin of nonreciprocal magnon that is distinct
from the Dzyaloshinskii-Moriya interaction in a honeycomb antiferromagnet. The
key ingredient is a symmetric anisotropic exchange interaction depending on the
bond direction, which results in a valley-type nonreciprocal magnon excitations
under the staggered antiferromagnetic ordering. Furthermore, we find this type
of nonreciprocal magnon exhibits a peculiar magnetic-field response; the
nonreciprocal direction can be manipulated by the in-plane rotating magnetic
field. The obtained results can be accounted for the emergence of the magnetic
toroidal multipoles.
