  The microscopic conditions for the emergence of nonreciprocal magnon
excitations in noncentrosymmetric magnets are theoretically investigated. We
show that asymmetric magnon excitations appear when a bond magnetic toroidal
dipole becomes active, which is defined as a parallel component between the
Dzyaloshinskii-Moriya vector and the averaged spin moments at the ends of the
bonds. Depending on magnetic structures accompanying the bond magnetic toroidal
dipoles, the higher-rank magnetic toroidal multipoles can also be activated in
a magnetic cluster, which describes various angle dependences of asymmetric
magnon excitations. We demonstrate a variety of asymmetric magnon excitations
for two magnetic systems in the noncentrosymmetric lattice structures; a
one-dimensional breathing chain under a uniform crystalline electric field and
a three-dimensional layered breathing kagome structure. We show that a bottom
shift of magnon bands occurs by the magnetic toroidal dipole and a valley
splitting occurs by the magnetic toroidal octupole under magnetic orderings.
