Asymmetric dispersion relation in spin-spiral structures

The asymmetric spin-wave dispersion relation observed recently [Zakeri et al., Phys. Rev. Lett. 104, 137203 (2010)], is explained within a quantum model consisting of an anisotropic Heisenberg coupling and the Dzyaloshinski-Moriya interaction (DMI). Applying a transformation of the spin operators into a representation without fixed quantization axis, a Green's function technique for finite temperatures allows to obtain the spin-wave dispersion for various magnetic spiral structures including cycloidal and conical ones. In case of vanishing anisotropy, the DMI drives an incommensurate antiferromagnetic alignment forming a cycloidal spin spiral. The spin-wave dispersion is symmetric. For nonzero anisotropy, the spins are aligned in a transverse conical spiral and the dispersion becomes asymmetric in agreement with the experimental observation. The asymmetry is reduced with increasing temperature.