We study the nature of melting of a two-dimensional (2D) Lennard-Jones solid using large-scale Monte Carlo simulation. We use systems of up to 102 400 particles to capture the decay of the correlation functions associated with translational order (TO) as well as the bond-orientational (BO) order. We study the role of dislocations and disclinations and their distribution functions. We computed the temperature dependence of the second moment of the TO parameter (${\Psi }_G$) as well as of the order parameter ${\Psi }_6$ associated with BO order. By applying finite-size scaling of these second moments, we determined the anomalous dimension critical exponents $\eta (T)$ and ${\eta }_6(T)$ associated with power-law decay of the ${\Psi }_G$ and ${\Psi }_6$ correlation functions. We also computed the temperature-dependent distribution of the order parameters ${\Psi }_G$ and ${\Psi }_6$ on the complex plane that supports a two-stage melting with a hexatic phase as an intermediate phase. From the correlation functions of ${\Psi }_G$ and ${\Psi }_6$, we extracted the corresponding temperature-dependent correlation lengths $\xi (T)$ and ${\xi }_6(T)$. The analysis of our results leads to a consistent picture strongly supporting a two-stage melting scenario as predicted by the Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY) theory where melting occurs via two continuous phase transitions, first from solid to a hexatic fluid at temperature $T_m$, and then from the hexatic fluid to an isotropic fluid at a critical temperature $T_i$. We find that $\xi (T)$ and ${\xi }_6(T)$ have a distinctly different temperature dependence, each diverging at different temperature, and that their finite-size scaling properties are consistent with the KTHNY theory. We also used the temperature dependence of $\eta$ and ${\eta }_6$ and their theoretical bounds to provide estimates for the critical temperatures $T_m$ and $T_i$, which can also be estimated using the Binder ratio. Our results are within error bars, the same as those extracted from the divergence of the correlation lengths.

• Received 28 March 2011

DOI:https://doi.org/10.1103/PhysRevB.83.214108