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

Abstract

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 $H_1=k_B{\mathrm{Th}}_1$ 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 $\Delta f(n,t,h_1)\approx n^{-2\mathrm{}}\left(\ln n\right)W_1(nt,n^{\frac{1}{2}}{h}_1)+n^{-2\mathrm{}}{W}_2(nt,n^{\frac{1}{2}}{h}_1)$, where $t=\frac{(T-T_c)}{T_c}$. The scaling functions $W_i\mathrm{}(x,y)\mathrm{}$ are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\to \infty$ at fixed $t\ne 0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order $e^{\frac{\mathrm{pna}}{\xi \mathrm{}\left(T\right)\mathrm{}}}$, where $a$ is the lattice spacing and $\xi \mathrm{}\left(T\right)\mathrm{}$ 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").