Systematic expansion in the order parameter for replica theory of the dynamical glass transition

TL;DR

This paper develops a systematic expansion around replica theory results for the dynamical glass transition, incorporating static three-body correlations and highlighting the limitations of static computations in capturing mode-coupling dynamics.

Contribution

It introduces a systematic expansion in the order parameter for replica theory, including the first correction involving three-body correlations, bridging static and dynamic approaches.

Findings

First correction involves static three-body correlations.

Higher order terms are quantitatively significant at the transition.

Static computations cannot fully recover mode-coupling kernels.

Abstract

It has been shown recently that predictions from Mode-Coupling Theory for the glass transition of hard-spheres become increasingly bad when dimensionality increases, whereas replica theory predicts a correct scaling. Nevertheless if one focuses on the regime around the dynamical transition in three dimensions, Mode-Coupling results are far more convincing than replica theory predictions. It seems thus necessary to reconcile the two theoretic approaches in order to obtain a theory that interpolates between low-dimensional, Mode-Coupling results, and "mean-field" results from replica theory. Even though quantitative results for the dynamical transition issued from replica theory are not accurate in low dimensions, two different approximation schemes --small cage expansion and replicated Hyper-Netted-Chain (RHNC)-- provide the correct qualitative picture for the transition, namely a discontinuous jump of a static order parameter from zero to a finite value. The purpose of this work is to develop a systematic expansion around the RHNC result in powers of the static order parameter, and to calculate the first correction in this expansion. Interestingly, this correction involves the static three-body correlations of the liquid. More importantly, we separately demonstrate that higher order terms in the expansion are quantitatively relevant at the transition, and that the usual mode-coupling kernel, involving two-body direct correlation functions of the liquid, cannot be recovered from static computations.