Diverging length scale and upper critical dimension in the Mode-Coupling Theory of the glass transition

TL;DR

This paper demonstrates that the Mode-Coupling Theory of the glass transition predicts critical phenomena with diverging length and time scales, but these predictions are mean-field and expected to change below the upper critical dimension of 6.

Contribution

It provides the first detailed analysis of the critical exponents and scaling forms in MCT, highlighting their mean-field nature and the role of the upper critical dimension.

Findings

Identifies diverging length and time scales near the glass transition.

Derives scaling exponents nu and z relating space and time scales.

Predicts that MCT results are mean-field and change below dimension 6.

Abstract

We show that the glass transition predicted by the Mode-Coupling Theory (MCT) is a critical phenomenon with a diverging length and time scale associated to the cooperativity of the dynamics. We obtain the scaling exponents nu and z that relate space and time scales to the distance from criticality, as well as the scaling form of the critical four-point correlation function. However, both these predictions and other well known MCT results are mean-field in nature and are thus expected to change below the upper critical dimension dc=6, as suggested by different forms of the Ginzburg criterion.