Surface tension, roughening, and lower critical dimension in the random-field Ising model

A continuum interface model is constructed to study the low-temperature properties of domain walls in the random-field Ising model (RFIM). The width of the domain wall and its surface tension are computed by three methods: Simple energy accounting, dimensional arguments, and approximate renormalization-group calculations. All methods yield a surface tension which is positive at sufficiently low temperature for small random fields, h, provided that the dimensionality $d>2$. The lower critical dimension of the RFIM is thus argued to be 2. While effects due to discreteness of a lattice are argued to alter some of the continuum results quantitatively, they do not change these central conclusions. For $d<2$ the ferromagnetic correlation length of the RFIM behaves like h^-2(2-d)^-1 as h→0.