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Journal of Physics A Mathematical and General

Nonuniversal critical behaviour in planar Ising models with extended, radially symmetric defects

R. Z. Bariev, Ingo Peschel

Published Jan 21, 1991• 16 citations
Publisher

Abstract

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.
DOI
10.1088/0305-4470/24/2/007

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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.