One-stage continuous melting transition in two dimensions.

Abstract

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 ${\ensuremath{\upsilon}}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.260\ifmmode\pm\else\textpm\fi{}0.005$; (2) the orientational order parameter drops discontinuously to zero at $\ensuremath{\upsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\upsilon}}_{c}$; (3) ${\ensuremath{\eta}}_{6}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.30\ifmmode\pm\else\textpm\fi{}0.05$ at $\ensuremath{\upsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\upsilon}}_{c}$; (4) there is consistency with $\ensuremath{\xi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{exp}({b/u}^{1/2})$ in the isotropic phase, where $u\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\upsilon}\ensuremath{-}{\ensuremath{\upsilon}}_{c}$ and $b\ensuremath{\simeq}0.8$. An intermediate phase in which the volume can vary by more than about 1% is ruled out.