| Several two dimensional systems with continuous symmetry have been explored using Monte Carlo computer simulation. The systems studied include the planar spin model, a novel hard spin model, the Lennard-Jones system at three densities and the hard disk model.; In the planar spin model our results show that the behavior of the susceptibility, spin-spin correlation function and the vortices were found to be consistent with the Kosterlitz-Thouless (KT) theory of vortex unbinding at the phase transition. We found k(,B)T(,c)/J = 0.89, where J is equal to the nearest neighbor coupling constant. The specific heat showed a peak at k(,B)T(,c)/J = 1.02, sharper than predicted by simple theory, which could be due to large clusters of vortices found near the transition.; The potential for the hard spin model is defined to be infinite if the cosine of the angular difference between a spin and any of its nearest neighbors is less than a specified parameter (beta). Otherwise the potential is zero. We find the susceptibility for this model diverges at (beta) (DBLTURN) -0.30 and there is a sharp change in the slope of the nearest neighbor correlation versus (beta) curve near (beta) = -0.45. Vortices also appear at (beta) = -0.45.; We find numerical agreement between the values obtained for the elastic constant in the Lennard-Jones system at densities (rho)(sigma)('2) = 0.888 and 0.856 and the Kosterlitz-Thouless-Halperin-Nelson-Young theory of melting. Here (sigma) is the hard core size parameter of the Lennard-Jones potential. Our results for other quantities could be consistent with either the Halperin-Nelson theory or a first order interpretation of melting. It is possible that if melting is second order in the thermodynamic limit, it might appear first order in the finite size finite time simulations which we are able to carry out. The transitions at a very high density, (rho)(sigma)('2) = 1.143, and in the hard disk system are more likely to be first order transitions. |
