Application of the Method of Two-Time Green's Function to the Spin-½ Heisenberg Ferromagnet

In the usual theory of the two-time Green's function, decoupling techniques are introduced in order to obtain a self-closed equation for the lowest order Green's function. These decouplings give neither the correct temperature dependence (the absence of a $T^3$ term) of magnetization nor the correct correlation function $〈{S_j}^{-}{S_j}^{-}{S_k}^{+}{S_l}^{+}〉$, which should be zero by definition. In this paper the equation of motion for the higher order Green's function is solved in the approximation which will lead to correct results up to terms of order $T^3$ in these quantities. Employing this higher order Green's function, it is found that the correction to the magnetization consists of a term of order $T^4$ and a divergent term and that the correlation function consists of a term of order $T^3$ and a divergent term. A method extracting a significant part from this divergent term is found by analyzing a system of two interacting ½ spins. By this method, it is shown that the $T^3$ term in the magnetization is completely eliminated and that the correlation function $〈{S_j}^{-}{S_j}^{-}{S_k}^{+}{S_l}^{+}〉$ vanishes identically.