One-Stage Continuous Melting Transition in Two Dimensions

Monte Carlo results are reported for melting of two-dimensional systems of $N$ hard disks for various values of $N$. Runs of up to 1$0^8$ Monte Carlo sweeps give the following equilibrium results: (1) there is a single second-order transition at volume ${\upsilon }_c=1.260\mathrm{}\pm 0.005$; (2) the orientational order parameter drops discontinuously to zero at $\upsilon ={\upsilon }_c$; (3) ${\eta }_6=0.30\mathrm{}\pm 0.05$ at $\upsilon ={\upsilon }_c$; (4) there is consistency with $\xi =\exp \left({b/u}^{1/2}\right)$ in the isotropic phase, where $u=\upsilon -{\upsilon }_c$ and $b\simeq 0.8$. An intermediate phase in which the volume can vary by more than about 1% is ruled out.