Using the Ginzburg–Landau–Wilson (GLW) Hamiltonian, we obtain, with the mean-field approximation, explicit expressions for the spin–spin correlation function χ(z,z′) of a film of thickness L above the phase transition temperature Tc and the spontaneous magnetization [Formula: see text] below Tc. The boundaries are treated using a temperature-independent extrapolation length Λ. From our results, we verify explicitly that for finite L, the critical indices associated with the spin–spin correlation functions and the surface magnetization are identical with those for the analogous two-dimensional system, for both the ordinary (Λ > 0) and surface (Λ < 0) transitions. Our model results nicely exhibit the fact that as long as L is finite, when the temperature T approaches sufficiently close to Tc, there is a crossover from behaviour characteristic of a single surface to two-dimensional behaviour. Within the one-loop, Hartree self-consistent field approximation, we study the effects of mode–mode coupling on the surface layer susceptibility in films of varying thicknesses. The singular behaviour obtained in the mean-field approximation is found to be completely removed in systems of finite thickness, the susceptibilities only exhibiting a finite cusp at the transition.
Exact Expressions of Spin-Spin Correlation Functions of the Two-Dimensional Rectangular Ising Model on a Finite Lattice