Real-space renormalisation of the smoothly inhomogeneous semi-infinite Ising model

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Abstract
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 y0, and a non-universal A-dependent surface magnetic index in the marginal case y=v-1.