Universal magnetic fluctuations in the two-dimensional XY model

Abstract

We discuss the probability distribution function for the magnetic order parameter M, in the low temperature phase of the two-dimensional XY model. In this phase the system is critical over the whole range of temperature. The thermally averaged value of the order parameter 〈M〉, which is zero in the thermodynamic limit, has abnormally large finite size corrections. An exact result, within a spin wave calculation gives 〈M〉 = (1/2N) T/8πJ, where J is the magnetic exchange constant and N the number of spins. We show, using Monte Carlo simulation, that the distribution function, Q(y - 〈y〉), y = T-1LT/4πJM, is an asymmetric universal function. Using a diagramatic technique, we show that the asymmetry comes from three-spin and higher correlations. If only two-spin correlations are considered, the distribution is Gaussian. However, as there are contributions from two-spin terms separated by all distances, the distribution remains broad and is consistent with a divergent susceptibility. © 1998 American Institute of Physics.