Critical Dynamical Heterogeneities Close to Continuous Second-Order Glass Transitions

We analyze, using inhomogeneous mode-coupling theory, the critical scaling behavior of the dynamical susceptibility at a distance $\epsilon$ from continuous second-order glass transitions. We find that the dynamical correlation length $\xi$ behaves generically as ${\epsilon }^{-1/3}$ and that the upper critical dimension is equal to six. More surprisingly, we find that $\xi$ grows with time as ${\ln }^{2}t$ exactly at criticality. All of these results suggest a deep analogy between the glassy behavior of attractive colloids or randomly pinned supercooled liquids and that of the random field Ising model.

Figure 1

Schematic presentation of the FP potential. The second minimum that develops below the transition point becomes flat as the usual MCT ($A_2$) transition point is approached from below. The two minima coalesce into one at the $A_3$ point. Inset: Schematic phase diagram obtained for random pinning glass transitions. In the temperature ($T$)-pinning fraction ($c$) plane the MCT transition line ends in an $A3$ critical point [3]. The two special directions mentioned in the text are shown.

Figure 2

Numerical results obtained by solving the schematic $F_{13}$ equations. ${\chi }_{{\mathbf{q}}_0=0}$ as a function of time approaching the $A3$ critical point, which is indeed found to rescale as $|{\epsilon }_{\perp }{|}^{-2/3}$ times a function of $|{\epsilon }_{\perp }{|}^{1/6}\times \ln t$. Inset: Scaling collapse, for different $q_0$ and $t$, of $q_0^2{\chi }_{{\mathbf{q}}_0}^c$ as a function of $q_0{\ln }^{2}t$ at criticality, as predicted from the theory.