Crossover between ordinary and normal transitions in two dimensional critical Ising films

We investigate two dimensional critical Ising films of width L with surface fields H-1=H-L in the crossover between ordinary (H-1=0) and normal (H-1=infinity) transitions. Using exact transfer-matrix diagonalization and density matrix renormalization-group (DMRG) methods, we calculate magnetization profiles m(z), the excess magnetization Gamma, and the analog of the solvation force f(solv) as functions of H-1 for several L. Scaling functions of the above quantities deviate substantially from their asymptotic forms at fixed points for a broad region of the scaling variable LH12 similar to L/l(1), where l(1) is the length induced by the surface field H-1. The scaling function for \f(solv)\ has a deep minimum near LH12 = 1, which is about one order of magnitude smaller than its value at both fixed points (the "Casimir" amplitude). For weak H-1 (l(1)>L) the magnetization profile has a maximum at the center of the film, and f(solv) decays much faster than L-2. For stronger H-1 (l < l(1) < L), the magnetization has two maxima at a distance similar to l(1) from the walls, and the solvation force decays much slower than L-2. For L much greater than l(1) the solvation force decays according to the universal power law f(solv) similar to L-2. The results of the approximate DMRG method show remarkable agreement with the exact ones. [S1053-651X(99)00209-3].