Monte Carlo Study of Two-Dimensional Melting and the Laplacian Roughening Model

The problem of two-dimensional (2d) melting has been studied using Monte Carlo (MC) computer simulation. These simulations included studies of a system of particles interacting with a Lennard-Jones (LJ) potential and of the Laplacian roughening model. As a new probe of the nature of the 2d melting transition, a MC study of the nearest-neighbor bond-angular order parameter on various length scales was performed. This probe was used to distinguish between a hexatic phase and a two-phase region. Our results for a low-density LJ system are compared to results for a hard-disk system. We conclude that the melting transition of both the LJ and hard-disk systems is first order in these finite simulations. No qualitative difference in behavior between the two systems was observed. MC simulations were also used to study the thermodynamic properties of the Laplacian roughening model. Nelson has shown that this model is related via a duality transformation to a gas of disclinations in a two-dimensional solid, and mimics the positional and orientational symmetries relevant to 2d melting. In our simulations thermodynamic functions varied continuously over the full temperature range and agreed with analytic results at low and high temperatures. Correlation functions exhibited three distinct regimes as a function of temperature and yielded strong evidence for the existence of an intermediate phase characterized by short-range positional order and quasi-long-range orientational order. The MC results were in quantitative agreement with the predictions of the Kosterlitz-Thouless-Halperin-Nelson-Young theory and established the existence of two successive continuous phase transitions for the Laplacian roughening model. The corresponding transition temperatures were obtained from the temperature dependence on the renormalized elastic constant K(,R) and the Frank constant K(,A). We have also performed an exploratory Monte Carlo Renormalization Group (MCRG) study of the discrete Gaussian model for interfacial roughening. Since the characteristics of this system are well known, it was chosen as a testing ground for the application of the MCRG method to a roughening model. We were able to observe the high temperature line of fixed points. The location of the transition temperature by MCRG was in good agreement with the results of a standard MC simulation of the model. It is hoped that this study will provide useful preparation for a MCRG study of the Laplacian roughening model.