On the fragility of the mean-field scenario of structural glasses for finite-dimensional disordered spin models

TL;DR

This paper examines the limitations of mean-field models for structural glasses in finite dimensions, highlighting their fragility to short-range fluctuations and questioning their applicability to real three-dimensional glass transitions.

Contribution

It clarifies why mean-field models fail in finite dimensions and proposes testing models against short-range fluctuations before detailed simulations or renormalization-group analyses.

Findings

Mean-field models are only predictive in very high dimensions.

Bethe approximation offers insights into finite-dimensional behavior.

Disordered spin models may not be suitable for 3D glass transition studies.

Abstract

At the mean-field level, on fully connected lattices, several disordered spin models have been shown to belong to the universality class of "structural glasses", with a "random first-order transition" (RFOT) characterized by a discontinuous jump of the order parameter and no latent heat. However, their behavior in finite dimensions is often drastically different, displaying either no glassiness at all or a conventional spin-glass transition. We clarify the physical reasons for this phenomenon and stress the unusual fragility of the RFOT to short-range fluctuations, associated e.g. with the mere existence of a finite number of neighbors. Accordingly, the solution of fully connected models is only predictive in very high dimension whereas, despite being also mean-field in character, the Bethe approximation provides valuable information on the behavior of finite-dimensional systems. We suggest that before embarking on a full-blown account of fluctuations on all scales through computer simulation or renormalization-group approach, models for structural glasses should first be tested for the effect of short-range fluctuations and we discuss ways to do it. Our results indicate that disordered spin models do not appear to pass the test and are therefore questionable models for investigating the glass transition in three dimensions. This also highlights how nontrivial is the first step of deriving an effective theory for the RFOT phenomenology from a rigorous integration over the short-range fluctuations.