Bounded and inhomogeneous Ising models. II. Specific-heat scaling function for a strip

The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of n parallel layers with free boundary conditions and a surface magnetic field H₁=kBTh₁ imposed on the last layer, is analyzed for large n in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form Δf(n,t,h₁)≈n^-2(lnn)W₁(nt,n1/2h₁)+n^-2W₂(nt,n1/2h₁), where t=(T-Tc)Tc. The scaling functions Wᵢ(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit n→∞ at fixed t≠0, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order e^pnaξ(T), where a is the lattice spacing and ξ(T) is the bulk correlation length; the value of the small rational constant p is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").