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Progress of Theoretical Physics

d'-Dimensional Defect in d-Dimensional Lattice. I: --Nonuniversal Local Critical Exponent in the Limit n-> --

R. Abe

Published Apr 1, 1981• 11 citations
Publisher

Abstract

The n-vector model in a d-dimensional lattice is studied where a d′-dimensional defect exists. The defect is defined in such a way that the spin length there is different from that in other parts. The pair correlation function in the defect space is discussed in the limit n→∞. It is shown that if d′=2 a critical exponent η• pertaining to the critical correlation is nonuniversal; it does depend on λ where λ is a parameter describing the strength of defect. The η′ is calculated up to order of λ.

DOI
10.1143/ptp.65.1237

Related papers

Higher Order Calculations for d-Dimensional Lattice with d'-Dimensional Defect

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By means of 1/n expansion, the n-vector model in a d-dimensional lattice is studied where a d′-dimensional defect exists. Previous theories which dealt with the n→∞ limit are extended up to the order λ/n (λ: a parameter specifying the strength of defect). The pair correlation function in the defect space is discussed for the three-dimensional lattice. A critical exponent η// pertaining to the critical correlation is calculated up to the same order.

Progress of Theoretical Physics•Feb 1, 1983•3 citations

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Progress of Theoretical Physics•Jan 1, 1984•1 citation

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Progress of Theoretical Physics•Aug 1, 1983•1 citation

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Local magnetisation of the n-vector model with a large-scale defect in the limit n to infinity

R. Z. Bariev, Ilyas Z. Ilaldinov

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.

Journal of Physics A Mathematical and General•Sep 21, 1989•2 citations
The n-vector model in a d-dimensional lattice is studied where a d′-dimensional defect exists. The defect is defined in such a way that the spin length there is different from that in other parts. The pair correlation function in the defect space is discussed in the limit n→∞. It is shown that if d′=2 a critical exponent η• pertaining to the critical correlation is nonuniversal; it does depend on λ where λ is a parameter describing the strength of defect. The η′ is calculated up to order of λ.
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 critical behavior of the ddd-dimensional nnn-vector model near an internal $d\ensuremath{-}1$ dimensional plane of weakened interactions in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ changes qualitatively at d=3d=3d=3. For d>3d>3d>3 the critical exponent ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$, which characterizes the correlation of spins in the defect plane, is universal, i.e., is independent of the strength of the interactions in the defect plane. We argue that ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ is nonuniversal for d=3d=3d=3 and calculate its dependence on the difference between the defect and bulk couplings to second order in perturbation theory. We show that the excess energy density at a distance zzz from the defect plane decays at the critical point to the bulk value as ${z}^{\ensuremath{-}2}$ for $d\ensuremath{\ge}3$ and faster than ${z}^{\ensuremath{-}2}$ for d<3d<3d<3. Near the free surface of a semi-infinite system one finds a ${z}^{\ensuremath{-}2}$ dependence for d>3d>3d>3 as well as for $d\ensuremath{\ge}3$.
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.