Blocking transitions in lattice spin models with directed kinetic constraints

We present a class of kinetic Ising models with a directed constraint, which exhibit a dynamocal phase transition at a critical up-spin concentration. At lower up-spin concentrations a non-ergodic state exists, for which the fraction of permanently blocked spins plays the role of an order parameter. The spin autocorrelation function has been calculated by Monte Carlo simulation for two particular models: One for the square lattice, with the constraint that the neighboring spins in the north and in the east are up, and one for a Cayley tree with branching ratio 2, where both neighbors above a given spin must be up. The transition is of second order, and the integrated autocorrelation time diverges on both sides of the transition. The models define a distribution of blocking lengths, which can be calculated analytically for the Cayley tree. We conjecture that this length distribution is related to the autocorrelation time by a dynamic critical exponent, which is numerically estimated for small lattices up to a length 11 to be about 5 and 2.7 for the two models, respectively.