Critical exponents of isotropic-hexatic phase transition in the hard-disk system

The hard-disk system is studied by observing the nonequilibrium relaxation behavior of a bond-orientational order parameter. The density dependence of characteristic relaxation time $\tau$ is estimated from the finite-time scaling analysis. The critical point between the fluid and the hexatic phase is refined to be $0.899\left(1\right)$ by assuming the divergence behavior of the Kosterlitz-Thouless transition. The value of the critical exponent $\eta$ is also studied by analyzing the fluctuation of the order parameter at the criticality and estimated as $\eta =0.25\left(2\right)$. These results are consistent with the prediction by the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

Figure 1

Relaxation of the bond-orientational order ${\varphi }_6\left(t\right)$ for various densities from $\rho =0.800$ to $0.890$ in log-log plot. Natural logarithms are used for all figures.

Figure 2

Scaling plot of bond-orientational order for all densities with appropriately chosen $\tau \left(ϵ\right)$ and $\lambda$ in log-log plot.

Figure 3

Relaxation times $\tau \left(ϵ\right)$ in units of $\tau$ at $\rho =0.888$ in semilog plot. The solid curve is the line fitted using scaling form (4) with ${\rho }_i=0.899\left(1\right)$, $a=0.876\left(2\right)$, and $b=1.63\left(1\right)\times {10}^{-3}$.

Figure 4

Time evolution of the bond-orientational order parameter ${\varphi }_6\left(t\right)$. After initial relaxation, ${\varphi }_6\left(t\right)$ shows power-law decay. From the asymptotic behavior ${\varphi }_6\sim t^{-\eta ∕2z}$, the value of $\eta ∕2z$ is determined to be $0.05\left(1\right)$.

Figure 5

The behavior of the fluctuation of the bond-orientational order parameter. After an initial relaxation, it shows good power-law behavior. From the asymptotic behavior (7), the value of $d∕z$ is determined to be $0.8\left(1\right)$. With $d=2$, one can obtain $z=2.5\left(2\right)$.