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Journal of Physics A Mathematical and General

Correlation functions in the two-dimensional Ising model with large-scale inhomogeneity caused by a line defect

R. Z. Bariev

Published May 7, 1989• 12 citations
Publisher

Abstract

Exact results are obtained for the local magnetisation and correlation functions of the two-dimensional Ising model with a line defect which produces large-scale inhomogeneity in its vicinity. It is shown that near the critical point the correlation functions exhibit a non-scaling behaviour.
DOI
10.1088/0305-4470/22/9/008

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We consider the critical spin–spin correlation function of the 2D Ising model with a line defect whose strength is an arbitrary function of the position. By using path integral techniques in the continuum description of this model in terms of fermion fields, we obtain an analytical expression for the correlator as a functional of the position-dependent coupling. Thus, our result provides one of the few analytical examples that allows one to illustrate the transit of a magnetic system from scaling to non-scaling behavior in a critical regime. We also show that the non-scaling behavior obtained for the spin correlator along a nonuniformly altered line of an Ising model remains unchanged in the Ashkin–Teller model.
We consider the critical spin–spin correlation function of the 2D Ising model with a line defect whose strength is an arbitrary function of the position. By using path integral techniques in the continuum description of this model in terms of fermion fields, we obtain an analytical expression for the correlator as a functional of the position-dependent coupling. Thus, our result provides one of the few analytical examples that allows one to illustrate the transit of a magnetic system from scaling to non-scaling behavior in a critical regime. We also show that the non-scaling behavior obtained for the spin correlator along a nonuniformly altered line of an Ising model remains unchanged in the Ashkin–Teller model.
The correlation functions of the three-dimensional n-vector model are 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 with respect to their bulk value. The deviations decay as - lambda /r with the distance r from the defect. It is shown that at the critical point the correlation function exhibits nonscaling behaviour if d'=2 and nonuniversal behaviour if d'=1. The deviation of a critical exponent eta is calculated up to order of lambda .
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.
We consider two-dimensional Ising models with randomly distributed ferromagnetic bonds and study the local critical behavior at defect lines by extensive Monte Carlo simulations. Both for ladder and chain type defects, non-universal critical behavior is observed: the critical exponent of the defect magnetization is found to be a continuous function of the strength of the defect coupling. Analyzing corresponding stability conditions, we obtain new evidence that the critical exponent ννν of the bulk correlation length of the random Ising model does not depend on dilution, i.e. ν=1ν=1ν=1.