Classical Heisenberg Magnet in Two Dimensions

The equilibrium properties of a square planar array of classical spins with near-neighbor Heisenberg interactions have been examined for arrays of up to 2025 spins with and without periodic boundary conditions. Equilibrium values of the root-mean-square magnetization $M_{\mathrm{rms}}$ were obtained by Monte Carlo calculations. Sample spin arrays whose energy agreed with the ensemble average at a given temperature were taken as characteristic of that temperature and were employed to obtain instantaneous correlation functions. The results are less clear than those reported previously for three-dimensional systems: The Monte Carlo calculations converged more slowly because of the lower connectivity of the lattice, and, unlike the three-dimensional case, the short-range order has a range as large, or larger, than the largest sample dimensions. The results are consistent with the observation of Mermin and Wagner that the system does not order ferromagnetically at finite temperatures, and lend some credence to the conjecture of Stanley and Kaplan on the existence of a special ordered state possessing "long-ranged short-range order."