Phase Transition in the Two-Dimensional Heisenberg Ferromagnet

We develop a Green-function theory to describe the thermodynamic behavior of a plane square lattice with spins of magnitude one-half located at the lattice sites interacting via a nearest-neighbor Heisenberg ferromagnetic coupling. Our approximation technique involves a decoupling of the hierarchy of Green-function equations similar in some respects to that found in the random-phase approximation (RPA) but improved to include spin correlations neglected in the RPA. Such an improvement is essential for the two-dimensional problem. Our theory predicts a phase transition at the temperature given by $k{T}_c=2J$, where $J$ is the exchange parameter. As $T$ approaches $T_c$ from above, the static susceptibility diverges as $\frac{1}{(T-T_c)}$. The spontaneous magnetization is zero at all nonzero temperatures, both above and below the critical point. Therefore, our theory is consistent with the existing rigorous proof of Mermin and Wagner that the spontaneous magnetization must be zero for $T\ne 0$, and displays the divergent susceptibility predicted by Stanley and Kaplan from an analysis of high-temperature expansions for related two-dimensional spin systems.