Melting transition in two dimensions: A finite-size scaling analysis of bond-orientational order in hard disks

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Abstract
We describe a general and efficient method, based on computer simulations and applicable to a general class of fluids, that allows us to determine (i) bounds on the transition densities of the melting transition that are valid in the thermodynamic limit and (ii) the order of the phase transition. The bond-orientational order parameter, its susceptibility, and the compressibility are measured simulataneously on many length scales, and the latter two quantities are extrapolated to the thermodynamic limit by application of the subblock analysis method of finite-size scaling. We include a detailed analysis, related to the subblock method, of the cross correlations of the fluctuations of the density and the order parameter. The behavior of the extrapolated order parameter susceptibility yields precise upper and lower bounds for the melting and freezing densities, respectively. We apply these techniques to the two-dimensional melting transition in large systems of 16 384 hard disks using canonical Monte Carlo computer simulations. The measured bond-orientational susceptibilities are found to be incompatible with predictions of the Halperin-Nelson-Young theory of two-dimensional melting. This and the behavior of the bond-orientational cumulant are two strong pieces of evidence of a first-order phase transition.

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