Ideal glass transition in a simple two-dimensional lattice model

We present a simple lattice model showing a glassy behavior. $R$ matrix analysis predicts critical termination of the supercooled fluid branch at density ${\rho }_g=0.1717$. This prediction is confirmed by dynamical numerical simulations, showing power-law divergences of relaxation time ${\tau }_{1/2}$, as well as the four-susceptibility ${\chi }_4$ peak’s location and height exactly at the predicted density. The power-law divergence of ${\chi }_4$ continues up to ${\chi }_4$ as high as ${10}^4$. Finite-size scaling study reveals the divergence of the correlation length accompanying the transition.

Figure 1

(Color online) $N3$ equation of state: $R$ matrix prediction, based on the first 23 Mayer cluster integrals (dashed line), Monte Carlo calculation on a $1000\times 1000$ lattice (solid line), and exact transfer-matrix calculation for a semi-infinite 25 site wide strip (symbols). The latter two methods provide equilibrium results, while the $R$ matrix extrapolates to the supercooled fluid branch. The agreement of the $R$ matrix results with the numerical methods is excellent throughout the fluid regime. Inset shows the diagonal $\left(B_n\right)$ and off-diagonal $\left(A_n\right)$ $R$ matrix elements, together with the fitted asymptotic form.

Figure 2

(Color online) Density-density autocorrelation (2), showing the typical glassy relaxation picture, on a logarithmic time scale. The inset shows the relaxation time ${\tau }_{1/2}$ vs density (symbols), which is well fitted by a power law (3) diverging at ${\rho }_g$.

Figure 3

(Color online) ${\chi }_4$ as a function of time showing the familiar peaks. Peaks’ heights ${\chi }_{\text{max}}$ and locations ${\tau }_4$ power law diverge as density approaches ${\rho }_g$ (inset) with critical exponents of 2.85 and 0.93, respectively.

Figure 4

(Color online) Finite-size analysis: ${\chi }_4$ peak height as a function of lattice linear size, for various densities. As the density approaches ${\rho }_g$, larger lattices are needed for converged results, attesting for a diverging length scale. For lattices smaller than the correlation length, ${\chi }_{\text{max}}$ is expected to grow like system’s size. A straight line with a slope of 2 is presented, to guide the eye.