Scaling and universality in glass transition

TL;DR

This paper develops a mean field scaling theory for glass transitions, connecting dynamical and static exponents, and demonstrating universality and scaling laws consistent with Mode Coupling Theory in facilitated spin models.

Contribution

It extends a cluster approach to describe the decay to the plateau and introduces a damage spreading mechanism, linking dynamical exponents to static bootstrap percolation exponents.

Findings

Scaling laws relate dynamical and static exponents.

Results match Mode Coupling Theory predictions.

Universal behavior emerges in mean field glass transition.

Abstract

Kinetic facilitated models and the Mode Coupling Theory (MCT) model B are within those systems known to exhibit a discontinuous dynamical transition with a two step relaxation. We consider a general scaling approach, within mean field theory, for such systems by considering the behavior of the density correlator <q(t)> and the dynamical susceptibility <q^2(t)> -<q(t)>^2. Focusing on the Fredrickson and Andersen (FA) facilitated spin model on the Bethe lattice, we extend a cluster approach that was previously developed for continuous glass transitions by Arenzon et al (Phys. Rev. E 90, 020301(R) (2014)) to describe the decay to the plateau, and consider a damage spreading mechanism to describe the departure from the plateau. We predict scaling laws, which relate dynamical exponents to the static exponents of mean field bootstrap percolation. The dynamical behavior and the scaling laws for both density correlator and dynamical susceptibility coincide with those predicted by MCT. These results explain the origin of scaling laws and the universal behavior associated with the glass transition in mean field, which is characterized by the divergence of the static length of the bootstrap percolation model with an upper critical dimension d_c=8.