Discontinuous interface depinning from a rough wall

Depinning of an interface from a random self-affine substrate with roughness exponent zeta(S) is studied in systems with short-range interactions. In two dimensions transfer matrix results show that for zeta(S) < 1/2 depinning falls in the universality class of the flat case. When zeta(S) exceeds the roughness (zeta(0) = 1/2) of the interface in the bulk, geometrical disorder becomes relevant and, moreover, depinning becomes discontinuous. The same unexpected scenario, and a precise location of the associated tricritical point, are obtained for a simplified hierarchical model. It is inferred that, in three dimensions, with zeta(S) = 0, depinning turns first order already for zeta(S) > 0. Thus critical wetting may be impossible to observe on rough substrates.