Abstract :
We studied the ferromagnetic Ising model on two-dimensional systems with rough boundaries and several thickness distributions. First, we considered very long strips with discretized Gaussian distributions of widths with mean 3 L 12. Systems with fixed interface width W and with increasing roughness (W Lβ, with 0 β 1) were analysed. We also considered strips with columns’ heights correlations, where the difference in the heights of neighbouring columns was 0 or 1, with 0 β 0.5. Free energies per spin, fL, and initial susceptibilities, χL, were calculated using transfer matrix techniques at the critical temperature of the two-dimensional Ising model. The scaling of fL and of the finite-size estimates of the ratio of exponents (γ/ν)L was analysed. In systems with small and slowly increasing W (β<0.2), correlated or not, extrapolations of fL and (γ/ν)L to L→∞ give accurate estimates of the critical parameters f∞ and γ/ν. In systems with rapidly increasing roughness, the deviations of those estimates from the exact values indicate the presence of large corrections in finite-size scaling relations. The comparison of systems with nearly the same β shows that spatial correlations have no systematic effect on corrections to scaling. It suggests the reliability of uncorrelated models or related approximate systems for the description of scaling properties of thin magnetic films and other low-dimensional structures.
