Using conformal mapping and finite-size scaling the authors determine the local magnetic exponent near a defect characterized by couplings of the form J(r)=J(1+A/r). In accordance with phenomenological considerations, they find that it depends continuously on the amplitude A, both for a defect in the interior and on the boundary.
We consider concentric circular defects in the two-dimensional Ising model, which are distributed according to a generalized Fredholm sequence, i.e. at exponentially increasing radii. This type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation. The critical exponent of the local magnetization is obtained through finite-size scaling, using a corner transfer matrix approach in the extreme anisotropic limit. It varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial Hilhorst-van Leeuwen model. Through a conformal mapping of the system onto a strip, the gap-exponent relation is shown to remain valid for such an aperiodic defect.
We consider a square-lattice Ising model with a smooth inhomogeneity in the bulk, where the couplings deviate from the bulk critical coupling by ${\mathit{Al}}^{\mathrm{\ensuremath{-}}1}$, l being the distance from the center of the inhomogeneity. Scaling exponents describing decay of correlations along the inhomogeneity have been determined exactly by conformal invariance and finite-size scaling. It is shown that the homogeneous model (A=0) with bulk Ising exponents separates the first-order transition regime (A>0) from the region of nonuniversal criticality (A0).
The local magnetisation of the three-dimensional n-vector model is studied near the large-scale d'-dimensional defect in the limit n to infinity . The model is characterised by the fact that the spin lengths close to the defect are changed as S r 2 =n(1- lambda /r) with the distance r from the defect. It is shown that near the critical point the local magnetisation exhibits a nonuniversal behaviour if d'=0, 1 and a nonscaling behaviour if d'=2. In the first case the critical exponents beta ' and nu ' are calculated up to the order of lambda and it is shown that these exponents satisfy the usual scaling law relations up to this order.