Non-Monotonic Dynamical Correlations Across The Glass Crossover

Abstract: The dramatic slowing down of structural relaxation in supercooled liquids is accompanied by the emergence of dynamic heterogeneity. A monotonically increasing dynamical correlation length, measured at the $α$-timescale, is one of the remarkable features of this phenomenon. Here we show that this picture is incomplete: the dynamical correlation length measured in the $β$-relaxation regime exhibits a striking non-monotonic temperature dependence, reaching a maximum near the mode-coupling crossover temperature $T_c$ and decreasing upon further cooling, even as local dynamical fluctuations continue to intensify. This behavior suggests a crossover from spatially extended, maximally cooperative motion near $T_c$ to increasingly compact and localized relaxation events below it. We demonstrate that this evolution is quantitatively captured by stochastic beta-relaxation theory, an extension of mode-coupling theory beyond mean-field that explicitly predicts an avoided dynamical transition in finite dimensions. Our results provide the first direct spatial evidence in favor of the avoided-transition picture of the mode-coupling crossover, and establish the peak of the $β$-regime correlation length as a robust indicator of the mode-coupling crossover.