Magnetization profile in the D=2 semi-infinite Ising model and crossover between ordinary and normal transition

We study the two-dimensional semi-infinite Ising model with a free surface at or near bulk criticality Special attention is paid to the influence of a boundary magnetic field h(1) on the surface-near regime and the crossover between the fixed points at h(1) = 0 and h(1) = infinity. Near the surface, a small h(1) causes a steeply increasing magnetization m(z) similar to z(3/8) log z as the distance z increases away from the surface. By means of a phenomenological scaling analysis, this phenomenon can be related to the well-known logarithmic dependence of the surface magnetization m(1) on h(1). Our analysis provides a deeper understanding of the existing exact results on m(zs) and relates the short-distance phenomena in d = 2 to those in higher dimensions. Both the results of the scaling analysis and the exact analytic profiles are corroborated by Monte Carlo simulations.