Self-Consistent Gaussian Approximation for Classical Spin Systems:   Thermodynamics
  The self-consistent Gaussian approximation (SCGA) for classical spin systems described by a completely anisotropic D-component vector model is proposed, which takes into account fluctuations of the molecular field and thus is a next step beyond the molecular field approximation. The SCGA is sensitive to the lattice dimension and structure and to the form of spin interactions and yields rather accurate values of the field-dependent magnetization m(H,T) and other thermodynamic functions in the whole plane (H,T) excluding the vicinity of the critical point (0,T_c), where the SCGA breaks down, showing a first-order phase transition. The values of T_c themselves can be determined in the SCGA with an accuracy better than 1% for actual 3-dimensional structures. At low and high temperatures the SCGA recovers the leading terms of the spin-wave theory, the low- and high-temperature series expansions, respectively. The accuracy of the SCGA increases with the increase of the spin dimension D, and in the limit D \to \infty the exact solution for the spherical model is recovered. 
D. A. Garanin