a Monte Carlo Study of the XY Model.

Monte Carlo simulation and analytical techniques are used to study the 2-D XY model. The relation between the partition functions of the Villain model and the Coulomb gas is used in combination with Villain's temperature relation (which relates the XY model and the Villain model) to obtain several new results. In particular, after verifying that at low temperatures a given configuration of vortices in the XY model and of charges in the Coulomb gas have the same energy, it is found that at intermediate temperatures the energy of vortices has decreased by approximately the factor 1/1.61. By Monte Carlo simulation it is found that this energy scale for vortices is necessary and sufficient for the sum of the spin-wave and vortex contributions to the specific heat to give the correct (total) specific heat. Villain's relations are used to transform the specific heat of the Coulomb gas, as calculated by other authors, into a result for the specific heat of the XY model, which agrees with the results of Monte Carlo simulations of the XY model. Monte Carlo simulation is also used to study the statistical mechanics of the vortex-vortex interaction. It is found that this interaction increases the density of vortices and also the specific heat for temperatures just less than that of the specific heat maximum (t(,M)). Just above t(,M) the density of vortices is found to decrease for increasing lattice size, suggesting the possibility of an anomaly in the specific heat at t(,M). The change at t(,M) in the form of the dependence of the vortex density on temperature is examined in terms of the vortex-vortex interaction energy. At t(,M), a maximum in this interaction energy is found to occur.