Nonreciprocal magnon propagation in a noncentrosymmetric ferromagnet LiFe$_{5}$O$_{8}$

In noncentrosymmetric materials, the relativistic effect extensively modifies the energy band of magnons as well as that of electrons. With use of microfabricated microwave antennae, we have demonstrated that the propagation of magnons with large momentum is nonreciprocal in a noncentrosymmetric ferromagnet LiFe$_{5}$O$_{8}$. The nonreciprocity is clearly explained by the effect of asymmetric magnon band originating from the relativistic Dzyaloshinskii-Moriya interaction. This result may pave a new path to designing magnonic device based on the relativistic band engineering.