The Exact Interface Model for Wetting in the Two-Dimensional Ising Model

We use exact methods to derive an interface model from an underlying microscopic model, i.e., the Ising model on a square lattice. At the wetting transition in the two-dimensional Ising model, the long Peierls contour (or interface) gets depinned from the substrate. Using exact transfer-matrix methods, we find that on sufficiently large length scales (i.e., length scales sufficiently larger than the bulk correlation length) the distribution of the long contour is given by a unique probability measure corresponding to a continuous interface model. The interface binding potential is a Dirac delta function with support on the substrate and, therefore, a distribution rather than a function. More precisely, critical wetting in the two-dimensional Ising model, viewed on length scales sufficiently larger than the bulk correlation length, is described by a reflected Brownian motion with a Dirac perturbation on the substrate so that exactly at the wetting transition the substrate is a perfectly reflecting surface; otherwise there exists a perturbation. A lattice solid-on-solid model was found to give identical results (albeit with modified parameters) on length scales sufficiently larger than the lattice spacing, thus demonstrating the universality of the continuous interface model.