Crossover scaling of apparent first-order wetting in two-dimensional systems with short-ranged forces

Recent analyses of wetting in the semi-infinite two-dimensional Ising model, extended to include both a surface coupling enhancement and a surface field, have shown that the wetting transition may be effectively first-order and that surprisingly the surface susceptibility develops a divergence described by an anomalous exponent with value γ_{11}^{eff}=3/2. We reproduce these results using an interfacial Hamiltonian model making a connection with previous studies of two-dimensional wetting, and we show that they follow from the simple crossover scaling of the singular contribution to the surface free-energy, which describes the change from apparent first-order to continuous (critical) wetting due to interfacial tunneling. The crossover scaling functions are calculated explicitly within both the strong-fluctuation and intermediate-fluctuation regimes, and they determine uniquely and more generally the value of γ_{11}^{eff}, which is nonuniversal for the latter regime. The location and the rounding of a line of pseudo-prewetting transitions occurring above the wetting temperature and off bulk coexistence, together with the crossover scaling of the parallel correlation length, are also discussed in detail.