Scaling treatment of the random field Ising model

Abstract

Analytic phenomenological scaling is carried out for the random field Ising model in general dimensions using a bar geometry. Domain wall configurations and their decorated profiles and associated wandering and other exponents $(ζ,γ,δ,μ)$ are obtained by free energy minimization. Scaling between different bar widths provides the renormalization group (RG) transformation. Its consequences are (1) criticality at $h=T=0$ in $d \leq 2$ with correlation length $ξ(h,T)$ diverging like $ξ(h,0) \propto h^{-2/(2-d)}$ for $d<2$ and $ξ(h,0) \propto \exp[1/(c_1γh^γ)]$ for $d=2$, where $c_1$ is a decoration constant; (2) criticality in $d = 2+ε$ dimensions at $T=0$, $h^{\ast}= (ε/2c_1)^{1/γ}$, where $ξ\propto [(s-s^{\ast})/s]^{-2ε/γ}$, $s \equiv h^γ$. Finite temperature generalizations are outlined. Numerical transfer matrix calculations and results from a ground state algorithm adapted for strips in $d=2$ confirm the ingredients which provide the RG description.