Computer simulations of the two-dimensional melting transition using hard disks

We present detailed Monte Carlo results for the two-dimensional melting transition of various systems up to $N=65\mathrm{}536$ hard disks. The simulations are performed in the $\mathrm{NVT}$ ensemble. In the isotropic phase the bond orientational correlation length ${\xi }_6$ and the susceptibility ${\chi }_6$ are measured and compared with the predictions of the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory. From the scaling relation of ${\xi }_6$ and ${\chi }_6$ we calculate the critical exponent ${\eta }_6.$ In the phase transition region we use finite-size scaling methods to locate the disclination binding transition point and compare the results with the values obtained from the behavior in the isotropic phase. Additionally, we measure the topological defect density, the pressure, and the distribution of the second moment of the local bond orientational order parameter. All results are in good agreement with the KTHNY theory, while a first-order phase transition with small correlation length and a one-stage continuous transition can be ruled out.