Size-stretched exponential relaxation in a model with arrested states.

We study the effect of a rapid quench to zero temperature in a model with competing interactions, evolving through conserved spin dynamics. In a certain regime of model parameters, we find that the model belongs to the broader class of kinetically constrained models, however, the dynamics is different from that of a glass. The system shows stretched exponential relaxation with the unusual feature that the relaxation time diverges as a power of the system size. Explicitly, we find that the spatial correlation function decays as exp(-2r/sqrt[L]) as a function of spatial separation r in a system with L sites in the steady state, while the temporal autocorrelation function follows exp[-(t/τ_{L})^{1/2}], where t is the time and τ_{L} proportional to L. In the coarsening regime, after time t_{w}, there are two growing length scales, namely L(t_{w})∼t_{w}^{1/2} and R(t_{w})∼t_{w}^{1/4}; the spatial correlation function decays as exp[-r/R(t_{w})]. Interestingly, the stretched exponential form of the autocorrelation function of a single typical sample in the steady state differs markedly from that averaged over an ensemble of initial conditions resulting from different quenches; the latter shows a slow power-law decay at large times.