Fate of mean-field dynamical decays in finite dimensions

Determine the fate of the dynamical power-law and logarithmic time-decays of correlation functions predicted by mean-field glassy theory when applied to realistic finite-dimensional systems (e.g., two- and three-dimensional spin glasses and supercooled liquids), clarifying whether these mean-field dynamical decays persist, are modified, or disappear due to finite-dimensional effects.

Background

The paper establishes, within a replicated Landau framework, that when the parameter exponent λ equals 1, glassy systems exhibit a logarithmic decay of dynamical correlations and provides a quantitative expression for the decay parameter μ in terms of static cumulants of the overlap. These results apply broadly to mean-field models such as multi-p-spin glasses and schematic Mode-Coupling Theory scenarios.

However, the authors emphasize that their results are inherently mean-field in nature. They note that for realistic finite-dimensional systems (two and three dimensions) the applicability of mean-field dynamical predictions is not guaranteed. While certain mean-field transitions (e.g., discontinuous dynamical arrest) are known to become crossovers in finite dimensions due to long-wavelength fluctuations, the specific dynamical signatures—power-law and logarithmic decays—lack a current theoretical understanding outside mean-field, motivating a precise determination of their behavior in finite dimensions.