Relaxation times of kinetically constrained spin models with glassy dynamics

We analyse the equilibrium dynamics of kinetically constrained spin systems which have been proposed as models for strong or fragile glasses and systems undergoing jamming transitions. We develop a novel multi-scale approach which allows us to connect the timescales to the typical size of the regions which have to be rearranged to create/destruct a particle. Thus we obtain exact results for the relaxation time τ as ρ → 1. For most of the models this regime was previously investigated only by numerical simulations. Depending on the choice of constraints, via our technique we obtain or we disprove the previously conjectured scalings. In particular, for the Fredrickson– Andersen and East models at any ρ < 1, we establish that the persistence and the spin–spin time autocorrelation functions decay exponentially. This excludes the stretched exponential relaxation which was derived by numerical simulations. Moreover for the East model we prove log τ = 1/(2 log 2)(log(1 − ρ))2 as ρ → 1, getting a sharp result for log τ which differs by a factor of 2 from the one conjectured in previous literature. For FA2f in d ≥ 2 and FA3f in d ≥ 3 we obtain the super-Arrhenius bounds exp (1 − ρ)−1 ≤ τ ≤ exp (1 − ρ)−5 and exp exp (1 − ρ)−1 ≤ τ ≤ exp exp (1 − ρ)−2, respectively. For FA1f in d = 1, 2. we rigorously prove the power law scalings recently derived in Jack et al (2006 Preprint cond-mat/0601529). Our techniques are flexible enough to allow a variety of constraints. In particular we can also deal with models displaying an ergodicity-breaking transition at ρc < 1, e.g. the North-East model. In this case we prove exponential decay to equilibrium in the whole ergodic regime.