Glassy dynamics in the asymmetrically constrained kinetic Ising chain

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 neighbor is up. We give details of our previous solution of the nonequilibrium 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 $\overline{d}\sim t^{T\ln 2},$ 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 time scales and length scales, we propose a model for the dynamics of “superdomains” which are bounded by up-spins that are frozen on long time scales. From this we deduce that the equilibrium spin correlation and persistence functions should exhibit identical scaling behavior for low T, decaying as $g\left(\tilde{t}\right).\mathrm{}$ The scaling variable is $\tilde{t}{=(t/t}_{*}{)}^{T\ln 2},$ giving strongly stretched behavior for low T. The scaling function $g\left(\cdot \right)$ decays faster than exponential, however, and in the limit $\overrightarrow{T}0$ at fixed $\tilde{t}$ reaches zero at a finite value of $\tilde{t}.$