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

Abstract

The random-field Ising model shows extreme critical slowdown that has been described by activated dynamic scaling: the characteristic time for the relaxation to equilibrium diverges exponentially with the correlation length, $\ln τ\sim ξ^ψ/T$ , with $ψ$ an \textit{a priori} unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions $d$ less than a critical value $d_{DR} \simeq 5.1$, also associated with dimensional-reduction breakdown, $ψ=θ$ with $θ$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For $d>d_{DR}$ on the other hand, $ψ=θ-2λ$ where $θ=2$ and $λ>0$ a new exponent. At the upper critical dimension $d=6$, $λ=1$ so that $ψ=0$, 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 1-$d$ systems.