Real-space renormalization-group analysis for finite ferromagnetic systems

The real-space renormalization-group transformation based on the cumulant expansion is modified for systems with free-surface boundary conditions. A discussion is given of the accuracy and limitations of the method and of the extension to higher orders in the cumulant expansion. Free energies and heat capacities of ferromagnetic strips ($n\times \infty$) and slabs ($n\times \infty \times \infty$) are determined over a wide temperature range. Exact results are obtained by a transfer matrix technique for strips with $n<~8$, and compared to the renormalization results. The critical or pseudocritical behavior is found to approach the bulk limit as $n\to \infty$ according to a simple power law $\left|T_c\mathrm{}\right(n)-T_c(\infty \left)\right|\propto n^{-\lambda }$, where the shift exponent $\lambda$ is equal to the reciprocal of the critical length exponent. Renormalization equations are used for the first time to calculate the heat capacity of the three-dimensional Ising model over a wide temperature range. In addition, the effect of surface anisotropy is considered for ferromagnetic strips, and procedures are outlined for obtaining analytic derivatives of the free energy with respect to the renormalized parameters.