Green's-function diagrammatic technique for complicated level systems. II. An application to the spin-1 Heisenberg ferromagnet with easy-axis single-ion anisotropy

The Green's-function diagrammatic expansion technique developed by the authors is applied to a spin-1 Heisenberg ferromagnet with easy-axis single-ion anisotropy. Inconsistency, which prevails in the previous random-phase-approximation equation-of-motion calculations, is examined and shown to be eliminated in the present calculation. Spin-wave energies, correlation functions, and magnetization are calculated to the zeroth order and first order in $\frac{1}{z}$. Critical temperature is determined (in fact, for a more general system which includes also the anisotropic exchange interaction) and compared with the values obtained by the high-temperature-expansion technique. The agreement is generally within a few percent (less than 1% for the Ising cases). At low temperatures, the second order in $\frac{1}{z}$ calculation is carried out. We show, in the temperature expansion of the magnetization, Dyson's $T^4$ correction to the first Born approximation, along with a series of terms led by $T^2{e}^{-\beta D}$ due to the single-ion anisotropy.