Renormalization-group approach to surface critical behavior in the Ising model

A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The $d=2\mathrm{}$ square lattice and the $d=3\mathrm{}$ bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for $d=3\mathrm{}$ are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\epsilon$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.