Scaling treatment of the random field Ising model

TL;DR

This paper develops a phenomenological scaling approach for the random field Ising model across various dimensions, deriving critical behavior and exponents through domain wall analysis and renormalization group transformations.

Contribution

It introduces a novel scaling framework for the RFIM using domain wall configurations and derives critical exponents and behaviors in different dimensions, supported by numerical calculations.

Findings

Criticality at zero temperature and field in low dimensions with diverging correlation length.

Exponential divergence of correlation length at the critical point in two dimensions.

Confirmation of the scaling approach through numerical transfer matrix and ground state calculations.

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 $d \leq 2$ 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.