Static replica approach to critical correlations in glassy systems

TL;DR

This paper develops a static replica field theory to analyze critical correlations in glassy systems, deriving physical quantities and a Ginzburg criterion to understand the validity of mean field approximations.

Contribution

It introduces a novel static replica approach to critical correlations in glasses, including a Ginzburg criterion and numerical estimates from liquid theory.

Findings

Derived correlation length and critical exponents.

Established a Ginzburg criterion for the glass transition.

Provided numerical estimates using Hypernetted Chain approximation.

Abstract

We discuss the slow relaxation phenomenon in glassy systems by means of replicas by constructing a static field theory approach to the problem. At the mean field level we study how criticality in the four point correlation functions arises because of the presence of soft modes and we derive an effective replica field theory for these critical fluctuations. By using this at the Gaussian level we obtain many physical quantities: the correlation length, the exponent parameter that controls the Mode-Coupling dynamical exponents for the two-point correlation functions, and the prefactor of the critical part of the four point correlation functions. Moreover we perform a one-loop computation in order to identify the region in which the mean field Gaussian approximation is valid. The result is a Ginzburg criterion for the glass transition. We define and compute in this way a proper Ginzburg number. Finally, we present numerical values of all these quantities obtained from the Hypernetted Chain approximation for the replicated liquid theory.