We compute the temperature-dependent spin-wave spectrum and the magnetization for a spin system using the unified decoupling procedure for the high-order Greens functions for the exchange coupling and anisotropy, both in the classical and quantum cas
e. Our approach allows us to establish a clear crossover between quantum-mechanical and classical methods by developing the classical analog of the quantum Greens function technique. The results are compared with the classical spectral density method and numerical modeling based on the stochastic Landau-Lifshitz equation and the Monte Carlo technique. As far as the critical temperature is concerned, there is a full agreement between the classical Greens functions technique and the classical spectral density method. However, the former method turns out to be more straightforward and more convenient than the latter because it avoids any emph{a priori} assumptions about the systems spectral density. The temperature-dependent exchange stiffness as a function of magnetization is investigated within different approaches.
