Random Field Ising-like effective theory of the glass transition II: Finite Dimensional Models

TL;DR

This paper develops an effective theory for overlap fluctuations in finite-dimensional glass-forming systems, incorporating point-to-set correlations via a random-field and random-bond Ising model, to better understand the glass transition.

Contribution

It introduces a variational low-temperature approximation that accounts for spatial correlations, extending mean-field models to finite dimensions in glass transition studies.

Findings

Effective theory maps to a random-field + random-bond Ising model.

Approximation scheme is validated on a fully connected model.

Discussion on implications for the existence of a finite-temperature glass transition.

Abstract

As in the preceding paper we aim at identifying the effective theory that describes the fluctuations of the local overlap with an equilibrium reference configuration close to a putative thermodynamic glass transition. We focus here on the case of finite-dimensional glass-forming systems, in particular supercooled liquids. The main difficulty for going beyond the mean-field treatment comes from the presence of diverging point-to-set spatial correlations. We introduce a variational low-temperature approximation scheme that allows us to account, at least in part, for the effect of these correlations. The outcome is an effective theory for the overlap fluctuations in terms of a random-field + random-bond Ising model with additional, power-law decaying, pair and multi-body interactions generated by the point-to-set correlations. This theory is much more tractable than the original problem. We check the robustness of the approximation scheme by applying it to a fully connected model already studied in the companion paper. We discuss the physical implications of this mapping for glass-forming liquids and the possibility it offers to determine the presence or not of a finite-temperature thermodynamic glass transition.