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Physical review. B, Condensed matter

Universal and nonuniversal critical behavior of the n -vector model with a defect plane in the limit n→∞

Erich Eisenriegler, Theodore W. Burkhardt

Published Mar 1, 1982• 17 citations
Publisher

Abstract

The local critical behavior of the $d$-dimensional $n$-vector model near an internal $d\ensuremath{-}1$ dimensional plane of weakened interactions in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ changes qualitatively at $d=3$. For $d>3$ 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 $d=3$ 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 $z$ 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 $d<3$. Near the free surface of a semi-infinite system one finds a ${z}^{\ensuremath{-}2}$ dependence for $d>3$ as well as for $d\ensuremath{\ge}3$.

DOI
10.1103/physrevb.25.3283

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The local critical behavior of the ddd-dimensional nnn-vector model near an internal d\ensuremath−1d\ensuremath{-}1d\ensuremath−1 dimensional plane of weakened interactions in the limit n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞ changes qualitatively at d=3d=3d=3. For d>3d>3d>3 the critical exponent \ensuremathη\ensuremath∥{\ensuremath{\eta}}_{\ensuremath{\parallel}}\ensuremathη\ensuremath∥​, 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 \ensuremathη\ensuremath∥{\ensuremath{\eta}}_{\ensuremath{\parallel}}\ensuremathη\ensuremath∥​ 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{z}^{\ensuremath{-}2}z\ensuremath−2 for d\ensuremath≥3d\ensuremath{\ge}3d\ensuremath≥3 and faster than z\ensuremath−2{z}^{\ensuremath{-}2}z\ensuremath−2 for d<3d<3d<3. Near the free surface of a semi-infinite system one finds a z\ensuremath−2{z}^{\ensuremath{-}2}z\ensuremath−2 dependence for d>3d>3d>3 as well as for d\ensuremath≥3d\ensuremath{\ge}3d\ensuremath≥3.
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 effects of irrelevant surface perturbations on the local critical behavior of ($4\ensuremath{-}\ensuremath{\epsilon}$)-dimensional nnn-vector models with a free surface or an internal defect plane are investigated with particular attention paid to the corrections to scaling induced by the surface term $\frac{1}{2}c\ensuremath{\int}{\mathrm{surface}}^{}{\ensuremath{\varphi}}^{2}$ near the ordinary transition. A systematic expansion in ${c}^{\ensuremath{-}1}$ is developed. In the case of a semi-infinite system, ${c}^{\ensuremath{-}1}$ couples to the square of the normal derivative of $\ensuremath{\varphi}$. The associated crossover exponent is ${\ensuremath{\Phi}}^{o}=\ensuremath{-}\ensuremath{\nu}$ to all orders in $\ensuremath{\epsilon}$. In the case of an infinite-space model with a defect plane, ${c}^{\ensuremath{-}1}$ couples to the squares of the normal derivatives on either side of the plane and to the product of the normal derivative on one side with the normal derivative on the other side. The latter operator represents perpendicular interactions across the defect plane, while the other two describe parallel interactions between spins on the same side of the plane. The associated crossover exponents are ${\ensuremath{\Phi}}_{\ensuremath{\parallel}}^{o}=\ensuremath{-}\ensuremath{\nu}$ and ${\ensuremath{\Phi}}_{\ensuremath{\perp}}^{o}={\ensuremath{\gamma}}_{11}^{o}$. A reanalysis of Bray and Moore's derivation of the (incorrect) relation ${\ensuremath{\Phi}}^{o}={\ensuremath{\gamma}}_{11}^{o}=\ensuremath{\nu}\ensuremath{-}1$ is presented which reveals where their arguments go wrong.
We discuss the universality classes of the local critical behavior of a bulk system near a free surface and near an internal plane of defects. A phase diagram and plausible renormalization-group flow properties for a general model which encompasses both cases are described qualitatively. Several of the crossover exponents are related to bulk critical exponents or semi-infinite surface critical exponents. The picture which is proposed accomodates systems with nonuniversal local critical exponents near a plane of defects, such as the two-dimensional Ising model near a defect line, in a natural way. Necessary conditions for nonuniversal local critical exponents and for logarithmic corrections are derived. The singularities of the energy densities near a free surface and near an internal defect plane on approaching the bulk critical temperature are analyzed.
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 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 study the local critical behavior near the free surface of a d-dimensional system with a smooth inhomogeneity induced by a defect of dimension d∗{\mathit{d}}^{\mathrm{*}}d∗\ensuremath{\le}d-1 located on the surface. The couplings deviate from their bulk critical value by \ensuremath{\lambda}${\mathit{r}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ where ${\mathit{r}}_{\mathrm{\ensuremath{\perp}}}$ is the distance to the center of the defect. Using a phenomenological approach, the change in the surface critical exponents is calculated to first order in the perturbation amplitude \ensuremath{\lambda} when the system is marginal, i.e., when the decay exponent of the perturbation \ensuremath{\alpha} is equal to the scaling dimension ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\varepsilon}}}^{0}$ of the bulk energy-density operator. Finally, the critical behavior near extended defects in the bulk is compared to the surface case.