  Relativistic spin-orbit interaction drastically modifies electronic band and
endows emergent functionalities. One of the example is the Rashba effect. In
noncentrosymmetric systems such as interface and polar materials, the
electronic band is spin-splitted depending on the momentum direction owing to
the spin-orbit interaction, which is useful for the electric manipulation of
spin current. Similar relativistic band-modification is also emergent for spin
wave (magnon) in magnetic materials. The asymmetric magnon band dispersion
induced by the Dzyaloshinskii-Moriya interaction, which is antisymmetric
exchange interaction originating from the spin-orbit interaction, is
theoretically expected, and experimentally observed recently in
noncentrosymmetric ferromagnets. Here, we demonstrate that the nonreciprocal
microwave response can be induced by the asymmetric magnon band in a
noncentrosymmetric ferrimagnet LiFe$_5$O$_8$. This result may pave a new path
to designing magnonic device based on the relativistic band engineering.
