Elementary excitations in anisotropic narrow-band magnetic semiconductors

The Hamiltonian of a narrow-band $s$-$d$ magnet with easy axis anisotropy is represented exactly in the form of the Hamiltonian of the degenerate Hubbard model. The diagram technique for Hubbard operators is used to find the spectrum of single-particle Fermi and Bose excitations at low temperatures and low carrier concentrations. The dependence of the spin-polaron band narrowing on the anisotropy is obtained. Two branches of magnon excitations are found; one determines the temperature corrections to the magnetization at small anisotropy, the other corrections at large anisotropy. It is shown that at $T=0$ and for sufficiently great anisotropy spin waves are subject to collisionless Landau damping, and their group velocity has a logarithmic singularity at the boundary of the damping region.