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

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

Figure 1

Dispersion relations, obtained by the CGF method, with wave vector (0,0,k) and without magnetic field.

Figure 2

Magnetization against reduced temperature. Comparison of (i) QGF for $S=5/2,30$, (ii) CGF, and (iii) classical MFT. We also show the low-temperature asymptote (45) and the asymptote near the critical temperature (53). The critical temperature ${\tau }_c$ is that of the method used for obtaining the corresponding magnetization curve and $\tau \equiv k_{B}T/J_0$.

Figure 3

Magnetization curves rendered by different decoupling schemes. The methods are compared to Monte Carlo method.

Figure 4

Magnetization obtained by CGF and CSD within RPA. The two methods are compared to Monte Carlo method. Inset: difference between CGF, CSD (with RPA decoupling) and MC.

Figure 5

Comparison between the CGF methods (by different decoupling schemes) and LLL approach.

Figure 6

Field dependence of the relative magnetization variation $\widetilde{\delta m}$, (left) from QGF with RPA decoupling and (right) from CGF [Callen (a) and RPA (b) decoupling] and MC.

Figure 7

Exchange stiffness against the magnetization for different decoupling schemes obtained by CGF.