Activated dynamic scaling in the random-field Ising model: A nonperturbative functional renormalization group approach

The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length ln tau similar to xi(psi)/T, with psi an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions d less than a critical value d(DR) similar or equal to 5.1, also associated with dimensional-reduction breakdown, psi = theta with theta the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For d > d(DR), on the other hand, psi = theta - 2 lambda, where theta = 2 and lambda > 0 an additional exponent. At the upper critical dimension d = 6, lambda = 1, so that psi = 0, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.