Criticality of the discrete N-vector ferromagnet in a cubic lattice with a free surface

We study a discrete N-component spin ferromagnet with Hamiltonian βH= −NK∑〈i,j〉(Si·Sj)−N2L∑〈i,j〉(Si·Sj)2 in a semi-infinite cubic lattice. The coupling constants at the surface, Ks and Ls, are allowed to be different from the bulk ones, KB and LB. Using a simple real-space renormalization group procedure we obtain the N-evolution (N real) of the phase diagram, including the N→0 and N→∞ limits. The thermal (v) and crossover (φ) critical exponents at various critical and multicritical points are calculated.