Localized model for systems with double-exchange coupling

Magnetic spins in crystals of mixed valence can simultaneously experience two kinds of coupling: the superexchange and the double exchange. The latter coupling, first invoked by Zener and further worked out by Anderson and Hasegawa, is mediated by additional electrons or holes introduced into the system. In the present paper an effective site-spins interaction Hamiltonian for double exchange is formulated. This Hamiltonian involves ascending powers of the bilinear interaction ${\overrightarrow{\mathrm{S}}}_i·{\overrightarrow{\mathrm{S}}}_j$. The highest power is determined by the value of the site spin. Thus, for spin-½ the double exchange interaction looks like the ordinary Heisenberg-type coupling. Spin-1 Hamiltonian contains also a biquadratic coupling. Spin-$\frac{3}{2}$ includes a bicubic interaction, spin-2 a biquartic one, etc. It is argued that a localized description of systems with the double exchange is usually sufficient. The phase diagrams are entirely different from the one predicted by the semiclassical (large-spin) band theory proposed by de Gennes. The critical concentrations of the carriers are evaluated at $T=0\mathrm{}$ K in the mean-field-theory approximation. An applied magnetic field is shown to have little influence on the strength of the double-exchange coupling. Spin configurations in the presence of the field are also discussed. Finally, a spin-wave theory for such systems is constructed.