Second-order Green's-function approach for the study of the sensitivity of the Curie temperature to single-ion anisotropy

This paper utilizes the second-order Green's-function formalism to treat a Heisenberg ferromagnet in the presence of single-ion crystal-field anisotropy represented by the Hamiltonian: $H=-J\Sigma {i,\delta }^{}\left({\overrightarrow{\mathrm{S}}}_i·{\overrightarrow{\mathrm{S}}}_{i+\delta }\right)-D\Sigma i^{}{\left(S_i^z\right)}^2$. Second-order equations of motion for the Green's functions $〈〈S_i^z; S_j^z〉〉$ and $〈〈{S_i}^{+}; {S_j}^{-}〉〉$ are developed and the higher-order Green's functions are decoupled in the zeroth-order approximation in which the interspin correlations are not taken into account rigorously. The spin-correlation functions are derived and are solved self-consistently in the limit of zero spontaneous magnetization. The Curie temperature is thus obtained. Calculations are restricted to a simple-cubic lattice and positive values of $D$ only, and the sensitivity of the Curie temperature to the single-ion anisotropy is critically examined for a spin-1 lattice. It is seen that the results agree very closely with those of the Green's-function diagram technique. The results are also compared with those of the first-order Green's-function theory using the random-phase approximations of Lines using the correlated-effective-field approximation, and of the molecular-field theory.