Modification of the classical Heisenberg helimagnet by weak uniaxial anisotropy and magnetic field

Classical ground state of the isotropic Heisenberg spin Hamiltonian on a primitive Bravais lattice is known to be a single-Q planar spin spiral. A uniaxial anisotropy and external magnetic field distort this structure by generating the higher-order Fourier harmonics at wave-vectors nQ in the spatial spin configuration. These features are not captured in the formalism based on the Luttinger-Tisza theorem, where the classical ground state energy is minimized under the ``weak'' condition on the length of the spins. We discuss why the correct solution is lost in that approach and present an alternative microscopic treatment of the problem. It allows to find the classical ground state for general Q for both easy-axis and easy-plane uniaxial second-order anisotropy, and for any orientation of the magnetic field, by treating the effect of anisotropy (but not the field) as a perturbation to the exchange structure. As a result, the classical ground state energy, the uniform magnetization, and the magnetic Bragg peak intensities that are measured in experiment, are calculated.