Extraordinary transition for the anisotropic Heisenberg ferromagnet on a semi-infinite lattice

We obtain bulk and surface order parameters of the spin-12 anisotropic Heisenberg ferromagnet in a semi-infinite (d = 3) lattice, as functions of the temperature. This model is described by the Hamiltonian −β H = Σ〈i,j〉Kij[(1 − Δij)(σixσjx + σiyσiy) + σizσjz], where the coupling constant Kij and the anisotropy Δij equal KB and ΔB for bulk interactions and KS and ΔS for surface interactions. Using real-space renormalization-group techniques, we discuss the (possible) discontinuity on the first (or second) derivative of the surface magnetization at the extraordinary transition, where the surface maintains its magnetization as the bulk disorders. This transition has been extensively studied in the Ising case (Δ = 1), due to a controversy on the continuity of the first derivative of the surface magnetization at this point.