Nonreciprocity of spin waves in noncollinear magnets due to the Dzyaloshinskii-Moriya interaction

Broken inversion symmetry in combination with the spin-orbit interaction generates a finite Dzyaloshinskii-Moriya interaction (DMI), which can induce noncollinear spin textures of chiral nature. The DMI is characterized by an interaction vector whose magnitude, direction and symmetries are crucial to determine the stability of various spin textures, such as skyrmions and spin spirals. The DMI can be measured from the nonreciprocity of spin waves in ferromagnets, which can be probed via inelastic scattering experiments. In a ferromagnet, the DMI can modify the spin-wave dispersion, moving its minimum away from the $\Gamma$ point. Spin waves propagating with opposite wavevectors are then characterized by different group velocities, energies and lifetimes, defining their nonreciprocity. Here, we address the case of complex spin textures, where the manifestation of DMI-induced chiral asymmetries remains to be explored. We discuss such nonreciprocal effects and propose ways of accessing the magnitude and direction of the DMI vectors in the context of spin-polarized or spin-resolved inelastic scattering experiments. We show that only when a periodic magnetic system has finite net magnetization, that is, when the vector sum of all magnetic moments is nonzero, can it present a total nonreciprocal spin-wave spectrum. However, even zero-net-magnetization systems, such as collinear antiferromagnets and cycloidal spin spirals, can have spin-wave modes that are individually nonreciprocal, while the total spectrum remains reciprocal.