Time scale separation and dynamic heterogeneity in the low temperature East model

TL;DR

This paper analyzes the East model's non-equilibrium dynamics at low temperatures, revealing how characteristic time scales diverge and demonstrating dynamic heterogeneity at mesoscopic scales, with precise relaxation time computations and scale separation insights.

Contribution

It provides the first rigorous proof of time scale divergence and dynamic heterogeneity in the East model at low temperatures, with new algorithms for relaxation time estimation.

Findings

Characteristic time scales diverge similarly as temperature decreases.

Dynamic heterogeneity manifests at mesoscopic length scales.

Relaxation times depend sharply on domain size, confirming heterogeneity.

Abstract

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbor is 1. We focus on the glassy effects caused by the kinetic constraint as $q\downarrow 0$, where $q$ is the equilibrium density of the 0's. In the physical literature this limit is equivalent to the zero temperature limit. We first prove that, for any given $L=O(1/q)$, the divergence as $q\downarrow 0$ of three basic characteristic time scales of the East process of length $L$ is the same. Then we examine the problem of dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale $L=O(q^{-\gamma})$, $\gamma<1$, we show that the characteristic time scale of two East processes of length $L$ and $\lambda L$ respectively are indeed separated by a factor $q^{-a}$, $a=a(\gamma)>0$, provided that $\lambda \geq 2$ is large enough (independent of $q$, $\lambda=2$ for $\gamma<1/2$). In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $111..10$, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. A key result for this part is a very precise computation of the relaxation time of the chain as a function of $(q,L)$, well beyond the current knowledge, which uses induction on length scales on one hand and a novel algorithmic lower bound on the other. Finally we show that no form of time scale separation occurs for $\gamma=1$, i.e. at the equilibrium scale $L=1/q$, contrary to what was assumed in the physical literature based on numerical simulations.