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We compute the temperature-dependent spin-wave spectrum and the magnetizationfor a spin system using the unified decoupling procedure for the high-orderGreen's functions for the exchange coupling and anisotropy, both in theclassical and quantum case. Our approach allows us to establish a clearcrossover between quantum-mechanical and classical methods by developing theclassical analog of the quantum Green's function technique. The results arecompared with the classical spectral density method and numerical modelingbased on the stochastic Landau-Lifshitz equation and the Monte Carlo technique.As far as the critical temperature is concerned, there is a full agreementbetween the classical Green's functions technique and the classical spectraldensity method. However, the former method turns out to be more straightforwardand more convenient than the latter because it avoids any \emph{a priori}assumptions about the system's spectral density. The temperature-dependentexchange stiffness as a function of magnetization is investigated withindifferent approaches.

Unified decoupling scheme for exchange and anisotropy contributions and temperature-dependent spectral properties of anisotropic spin systems