Some remarks on Kadanoff's variational approximations for renormalization-group transformations

The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher $n$-vector model in $4-\epsilon$ dimensions and the $d$-dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\epsilon$ for the Wilson-Fisher model and are given for the $d=2,3, \mathrm{and} 4$ dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.