Spin-correlation function in the two-dimensionalXYmodel

We study the two-spin correlation function in the high-temperature phase of the two-dimensional XY model. We derive a roughening representation of this function in the generalized Villain model, which is useful for high-temperature analysis. In the ordinary Villain model, we use duality to show that the correlation function is identical at high temperatures to the spin-correlation function in the two-dimensional Ising model. We use this result, together with spin-wave analysis, to perform a Nelson-Rudnick matching calculation of the spin-correlation function in the ordinary Villain model, thereby obtaining an approximate expression for this function at all length scales and temperatures above $T_c$.