A proper lower bound approximation to the true free energy per site of an Ising model with infinite-range interaction is obtained by the Migdal-Kadanoff real space recursion formulae. The results for the critical behaviour are confirmed by mean field analysis.
Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am -y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v -1 , a surface magnetic phase at the bulk critical temperature for y<v -1 , A>0, and a non-universal A-dependent surface magnetic index in the marginal case y=v -1 .
The Hamiltonian H=-J 2 Sigma (ij) sigma i sigma j -(J 4 /N) Sigma (ij)(kl) sigma i sigma j sigma k sigma l where the first term is a nearest-neighbour pair interaction and the second term is an infinite-range pair-pair interaction, is proposed as a model of an Ising magnet with lattice coupling. The free energy is reduced to that of a rigid Ising model on the same lattice but with a modified pair interaction. In two and three dimensions, the phase transition is found to be first order for all J 4 . The relationship between this result and recent renormalization group calculations is discussed.