Analytical approximations for the hierarchically constrained kinetic Ising chain

The hierarchically constrained kinetic Ising model in one dimension is reviewed, and the results of several analytical approaches to the model are presented. Two standard approximation schemes, an effective-medium approximation and a mode-coupling approximation, are shown to fail. A new class of approximations, termed cluster approximations, is better suited for the model. It yields good results for the spin autocorrelation function, and also elucidates important general properties of the model—its connection with defect-diffusion models and the asymptotic long-time behavior of the autocorrelation function.