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

TL;DR

This paper investigates the activated dynamic scaling in the random-field Ising model using a nonperturbative functional renormalization group, revealing different scaling behaviors below and above a critical dimension, with implications for understanding critical slowdown.

Contribution

It provides a nonperturbative RG analysis showing how the barrier exponent relates to other critical exponents and identifies a new regime above a critical dimension with distinct scaling behavior.

Findings

For d < 5.1, the barrier exponent equals the temperature exponent.

For d > 5.1, the barrier exponent differs from the temperature exponent, involving a new exponent.

At d=6, the model exhibits conventional scaling with zero barrier exponent.

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.