Goldstone mode in the conical phase of helical magnets

We investigate theoretically the nature of the Goldstone mode in the conical phase of helical magnets such as MnSi. A Dzyaloshinsky-Moriya term in the action leads to helical order in the ground state, characterized by a pitch vector ${\vec q}$ [1]. The Goldstone mode in the helical phase, the helimagnon, is known to have an anisotropic dispersion relation of the form $\Omega^2 \propto k_z^2 + k_{\perp} ^4/q^2$, analogous to smectic or cholesteric liquid crystals [2]. In the presence of a homogeneous external magnetic field $H$ the helix is superimposed by a homogeneous magnetization, which leads to a conical phase [3]. The Goldstone mode in the latter is found to be a modified helimagnon, with a dispersion relation of the structure $\Omega^2 \propto \Omega_0^2 + H^2 k_{\perp}^2$. The additional term $\propto H^2 k_{\perp}^2$ is a result of the magnetic field breaking the rotational symmetry. In addition, there are remnants of ferromagnetic magnons with masses $\propto H^2$. \\ $[1]$ P. Bak and M.H. Jensen, J. Phys. C 13, L881 (1980). \\ $[2]$ D. Belitz, T.R. Kirkpatrick, and A. Rosch, Phys. Rev. B 73, 054431 (2006). \\ $[3]$ Y. Ishikawa, G. Shirane, J.A. Tarvin, and M. Kohgi, Phys. Rev. B 16, 4956 (1977).