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 $(\zeta ,\gamma ,\delta ,\mu )$ 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 $d\le 2$ with correlation length $\xi (h,T)$ diverging like $\xi (h,0)\propto h^{-2/(2-d)}$ for $d<2$ and $\xi (h,0)\propto \exp [1/(c_1\gamma h^{\gamma })]$ for $d=2$, where $c_1$ is a decoration constant; (2) criticality in $d=2+\epsilon$ dimensions at $T=0$, $h^{\ast }=(\epsilon /2c_1{)}^{1/\gamma }$, where $\xi \propto [(s-s^{\ast })/s{]}^{-2\epsilon /\gamma }$, $s\equiv h^{\gamma }$. 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.

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