Glassy dynamics in the East model

TL;DR

This paper investigates the glassy dynamics of the East model, revealing anomalous coarsening, divergence of equilibration times, and proposing a scaling model for equilibrium behavior at low temperatures.

Contribution

It provides a detailed analysis of non-equilibrium and equilibrium dynamics in the East model, introducing a scaling hypothesis and linking to the paste-all coarsening model.

Findings

Anomalous coarsening of down-spin domains with size ~ t^{T ln 2}

Divergence of equilibration times as t_* = exp(1/T^2 ln 2)

Scaling behavior of spin correlation and persistence functions at low T

Abstract

We study the dynamics of the East model, comprising a chain of uncoupled spins in a downward-pointing field. Glassy effects arise at low temperatures $T$ from the kinetic constraint that spins can only flip if their left neighbour is up. We give details of our previous solution of the non-equilibrium coarsening dynamics after a quench to low $T$ (Phys. Rev.\ Lett. 83:3238, 1999), including the anomalous coarsening of down-spin domains with typical size $\bar{d} \sim t^{T \ln 2}$, and the pronounced `fragile glass'-divergence of equilibration times as $t_*=\exp(1/T^2\ln 2)$. We also link the model to the paste-all coarsening model, defining a family of interpolating models that all have the same scaling distribution of domain sizes. We then proceed to the problem of equilibrium dynamics at low $T$. Based on a scaling hypothesis for the relation between timescales and lengthscales, we propose a model for the dynamics of `superdomains' which are bounded by up-spins that are frozen on long timescales. From this we deduce that the equilibrium spin correlation and persistence functions should exhibit identical scaling behaviour for low $T$, decaying as $g(\tilde{t})$. The scaling variable is $\tilde{t}=(t/t_*)^{T\ln 2}$, giving strongly stretched behaviour for low $T$. The scaling function $g(\cdot)$ decays faster than exponential, however, and in the limit $T\to 0$ at fixed $\tilde{t}$ reaches zero at a {\em finite} value of $\tilde{t}$.