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Journal of Physics A Mathematical and General

Renormalisation group calculations of the critical exponents of the n-vector model with a free surface

J Reeve, Anthony John Guttmann

Published Dec 1, 1981• 28 citations
Publisher

Abstract

The correlation function critical exponents eta /sub /// and eta perpendicular to of the semi-infinite phi 4 model with O(n) symmetry have been calculated to order epsilon 2 for the ordinary and special transitions. The results for the ordinary transition are inconsistent with the relation gamma 11 = nu -1 proposed by Bray and Moore (1977). Comparisons with series expansion results in two and three dimensions are presented for n=0 and n=1.

DOI
10.1088/0305-4470/14/12/028

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The correlation function critical exponents eta /sub /// and eta perpendicular to of the semi-infinite phi 4 model with O(n) symmetry have been calculated to order epsilon 2 for the ordinary and special transitions. The results for the ordinary transition are inconsistent with the relation gamma 11 = nu -1 proposed by Bray and Moore (1977). Comparisons with series expansion results in two and three dimensions are presented for n=0 and n=1.
The critical behavior of a semi-infinite nnn-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions near the special transition is studied. The renormalization of the relevant surface operators is discussed and explicitly carried out in two-loop order. It is found that the crossover exponent ${\ensuremath{\varphi}}_{s}$ related to the square of the order parameter in the surface does not satisfy the relation ${\ensuremath{\varphi}}_{s}=1\ensuremath{-}\ensuremath{\nu}$ due to Bray and Moore.
The 1/n expansion method is developed in the study of the critical behavior of the n-vector model with a free surface. The method is applied at the bulk critical temperature (T = Tc) in the absence of the surface ordering. The surface critical exponents η;// and η;⊥ are identified up to order 1/n for general spatial dimension d: 2 ≪ d ≪ 4 for the ordinary transition and 3 ≪ d ≪4 for the special transition. The results are compared with several scaling relations, other theories and some experiments.
Correlation-function exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ appropriate to the free-surface problem have been obtained by renormalization-group calculation to order ${\ensuremath{\epsilon}}^{2}$. By using the scaling relations ${\ensuremath{\gamma}}_{1}=\ensuremath{\nu}(2\ensuremath{-}{\ensuremath{\eta}}_{\ensuremath{\perp}})$ and ${\ensuremath{\gamma}}_{11}=\ensuremath{\nu}(1\ensuremath{-}{\ensuremath{\eta}}_{\ensuremath{\parallel}})$, expansions for ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{11}$ are obtained. These expansions are in agreement with the surface scaling relation $2{\ensuremath{\gamma}}_{1}\ensuremath{-}{\ensuremath{\gamma}}_{11}=\ensuremath{\gamma}+\ensuremath{\nu}$, but disagree with the relation ${\ensuremath{\gamma}}_{11}=\ensuremath{\nu}\ensuremath{-}1$ due to Bray and Moore.
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.
The critical behavior at the extraordinary transition of classical n-vector model on semi-infinite lattice is analyzed in the 1/n expansion scheme for arbitrary spatial dimensions 2≪d≪4. Due to pre-existing surface orderings, the analysis at the extraordinary transition is somewhat analogous to that of bulk systems at T≪Tc. We show that the transverse correlation function has no anomolous dimension caused by the surface, i.e., η//T=d and η⊥T=(d+η)/2 to all orders in 1/n, where η is the bulk exponent. On the other hand, for the longitudinal correlation function, the relations η//L=d+2 and η⊥L=(d+2)/2 are shown to hold in the large n limit.