Percolation approach to glassy dynamics with continuously broken ergodicity

We show that the relaxation dynamics near a glass transition with continuous ergodicity breaking can be endowed with a geometric interpretation based on percolation theory. At the mean-field level this approach is consistent with the mode-coupling theory (MCT) of type-A liquid-glass transitions and allows one to disentangle the universal and nonuniversal contributions to MCT relaxation exponents. Scaling predictions for the time correlation function are successfully tested in the ${\mathsf{F}}_{12}$ schematic model and facilitated spin systems on a Bethe lattice. Our approach immediately suggests the extension of MCT scaling laws to finite spatial dimensions and yields predictions for dynamic relaxation exponents below an upper critical dimension of 6.

Figure 1

Nonarrested part of correlator $\Delta \varphi (t,\epsilon )$ vs rescaled time $t/{\tau }_{\epsilon }$ for $\lambda =0$. Solid lines are the numerical solution of the MCT schematic model, Eqs. (1) and (2), above ($\epsilon >0$) and below ($\epsilon <0$) the transition line. The dotted line is the stretched relaxation regime Eq. (7) obtained by expanding the MCT exact solution to the leading order in $t/{\tau }_{\epsilon }$. The dashed line represents the late stage exponential decay.

Figure 2

Nonarrested part of correlator $\Delta \varphi (t,\epsilon )$ in scaling form (top) and in natural units (bottom) for $\lambda =0.4$. Solid lines are the exact solution of MCT schematic model, Eqs. (1) and (2). The dotted line is the stretched relaxation regime Eq. (7) with exponents determined according to percolation predictions Eqs. (12, 13, 14) as $x=1.186$, $c=y=0.457$, $z=2.372$, and $a=0.422$. The dashed line represents the late stage exponential decay.

Figure 3

Rescaled persistence function $\varphi (t,\rho )$ for a facilitated spin model on a Bethe lattice with order parameter critical exponent $\beta =2$ and critical density ${\rho }_{\mathrm{c}}=1/3$. Relaxation time ${\tau }_{\rho }$ is computed according to Eq. (17) and $\delta \rho =1-\rho /{\rho }_{\mathrm{c}}$. The dotted line is the stretched relaxation regime Eq. (7) with exponents determined from the measure of critical relaxation exponent $a$ and Eqs. (8) and (9). The dashed line represents the late stage exponential decay.