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, $\ln \tau \sim {\xi }^{\psi }/T$ , with $\psi$ 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, $\psi =\theta$ 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 $\lambda >0$ a new 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 1-$d$ systems.

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• Critical behavior of the random-field Ising model with long-range interactions in one dimension
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