Scaling treatment of the random-field Ising model

Analytic phenomenological scaling is carried out for the random field Ising model in general dimensions d 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 i) criticality at h = T = 0 in d less than or equal to 2 with correlation length xi(h,T) diverging like xi(h,0) alpha h(-2/(2-d)) for d < 2 and xi(h,0) alpha exp[1/(c(1) gamma h(gamma))] for d = 2, where c(1) is a decoration constant; ii) criticality in d = 2 + epsilon dimensions at T = 0, h* = (epsilon/2c(1))(1/gamma), where xi alpha [(s - s*)/s](-2 epsilon/gamma), s drop 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.