Domain scaling and marginality breaking in the random field Ising model

TL;DR

This paper develops a scaling theory for the random field Ising model in various dimensions, showing how wall roughening affects marginality and correlation lengths, supported by numerical simulations.

Contribution

It introduces a new scaling description incorporating wall roughening effects that remove marginality in the 2D case, with detailed exponents and numerical validation.

Findings

Wall roughening removes marginality in 2D RFIM.

Correlation length in 2D scales as an exponential of the inverse field.

Numerical techniques confirm theoretical predictions for strips up to width 11.

Abstract

A scaling description is obtained for the $d$--dimensional random field Ising model from domains in a bar geometry. Wall roughening removes the marginality of the $d=2$ case, giving the $T=0$ correlation length $\xi \sim \exp\left(A h^{-\gamma}\right)$ in $d=2$, and for $d=2+\epsilon$ power law behaviour with $\nu = 2/\epsilon \gamma$, $h^\star \sim \epsilon^{1/\gamma}$. Here, $\gamma = 2,4/3$ (lattice, continuum) is one of four rough wall exponents provided by the theory. The analysis is substantiated by three different numerical techniques (transfer matrix, Monte Carlo, ground state algorithm). These provide for strips up to width $L=11$ basic ingredients of the theory, namely free energy, domain size, and roughening data and exponents.