Simulating the Melting Transition of Helium in Two Dimensions

We study the melting behavior of $^4$He in two dimensions with the path integral Monte Carlo method. Systems of helium atoms are simulated in a periodic cell designed to accommodate a triangular solid. We calculate the translational and orientational order parameters, as well as the defect fraction. Defects are defined as atoms with more or less than six neighbors; the nearest neighbor network is found through Delaunay triangulation. Two dimensional melting is a defect-mediated phase transition, thus, defects will proliferate as the solid is melted. Additionally, melting is expected to occur via a two-stage process, with transitions for both translational and orientational order. At high number density (0.0846 \AA$^{-2}$), we have seen a single transition (within the accuracy of our simulations). We are currently working to observe the melting transition at lower densities.