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Journal of Physics C Solid State Physics

Surface magnetisation profile in the critical region by the epsilon -expansion

C A Wilson

Published Feb 20, 1980• 13 citations
Publisher

Abstract

The Wilson perturbation theory technique is used to investigate the spontaneous magnetisation profile in the critical region for the ordinary transition of a scalar (n=1) semi-infinite model. The behaviour in the vicinity of the surface is explicitly determined and the surface exponent beta 1 identified to be 1- epsilon /6 to first order in epsilon =4-d, in agreement with the scaling prediction of Bray and Moore (1977).

DOI
10.1088/0022-3719/13/5/023

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The Wilson perturbation theory technique is used to investigate the spontaneous magnetisation profile in the critical region for the ordinary transition of a scalar (n=1) semi-infinite model. The behaviour in the vicinity of the surface is explicitly determined and the surface exponent beta 1 identified to be 1- epsilon /6 to first order in epsilon =4-d, in agreement with the scaling prediction of Bray and Moore (1977).
The critical behaviour of a semi-infinite system with O(n) spin symmetry is investigated using four techniques: (i) mean-field theory, (ii) exact solution for n= infinity , (iii) the epsilon expansion, and (iv) scaling arguments. The surface equation of state is computed using mean-field theory, and the four phase transitions defined by Lubensky and Rubin, the 'ordinary', 'surface', 'extraordinary', and ' lambda = infinity ' ('special') transitions, are identified. The first three may be observed in the bulk system by adding a suitable 'surface' perturbation which destroys translational invariance. Scaling arguments give their critical exponents exactly in terms of bulk exponents. Critical exponents for the ' lambda = infinity ' transition are determined for n= infinity and also to O( epsilon ). Spin-spin correlation functions at T=T c for the 'ordinary', 'extraordinary' and ' lambda = infinity ' transitions are calculated exactly for n= infinity .
The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents in a semi-infinite Ising model to first order in $\ensuremath{\epsilon}$. It is found that potentials renormalize just as in the bulk, and all surface information is contained in the wave-function renormalization. $\ensuremath{\nu}$ is $\frac{1}{2}+\frac{1}{12}\ensuremath{\epsilon}$ and $\ensuremath{\eta}$ is 0, just as in infinite systems. ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ are, respectively, $1\ensuremath{-}\frac{1}{6}\ensuremath{\epsilon}$ and $2\ensuremath{-}\frac{1}{3}\ensuremath{\epsilon}$ and the surface gap exponent ${\ensuremath{\Delta}}_{1}$ is $\frac{1}{2}\ensuremath{-}\frac{1}{12}\ensuremath{\epsilon}$ if scaling is assumed.
Using a recently introduced renormalisation group method, the authors study the behaviour of the spontaneous surface and bulk magnetisations as functions of the temperature for an Ising ferromagnet in a semi-infinite cubic lattice for various J S /J B ratios (J S and J B respectively are the surface and bulk coupling constants). In particular, they study the extraordinary transition where the surface maintains its magnetisation as the bulk disorders: the authors find a discontinuity in the first derivative of the surface magnetisation at the bulk transition temperature. The criticality of the system (universality classes, critical exponents and amplitudes) is discussed as well.
The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for nnn-dimensional classical spins on a semi-infinite lattice with surface exchange such that the extrapolation length is positive. It is found that to first order in $\ensuremath{\epsilon}$, all surface exponents can be calculated from bulk exponents and a single surface exponent, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\frac{(\frac{1}{2})\ensuremath{\epsilon}(n+2)}{(n+8)}$, describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ introduced by Binder and Hohenberg are, respectively, $1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ and $2(1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}})$. A form for the fixed-point spin correlation valid for all dimensions containing only the exponents $\ensuremath{\eta}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from $\ensuremath{\eta}$, $\ensuremath{\nu}$, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$ if scaling is assumed.
We calculate the surface critical exponents of the ordinary transition occurring in semi-infinite, quenched dilute Ising-like systems. This is done by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation as well as in 4-epsilon dimensions. At d=4-epsilon we extend, up to the next-to-leading order, the previous first-order results of the square root of epsilon expansion by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. In both cases numerical estimates for surface exponents are computed using Padé approximants extrapolating the perturbation theory expansions. The obtained results indicate that the critical behavior of semi-infinite systems with quenched bulk disorder is characterized by the new set of surface critical exponents.
The authors consider m two-dimensional semi-infinite planes of Ising spins joined together through surface spins and study the critical behaviour near to the junction. The m=0 limit of the model-according to the replica trick-corresponds to the semi-infinite Ising model in the presence of a random surface field (RSFI). Using conformal mapping, second-order perturbation expansion around the weak- and strong-coupled planes limits and a differential renormalization group, they show that the surface critical behaviour of the RSFI model is described by Ising critical exponents with logarithmic corrections to scaling, while at multiple junctions (m>2) the transition is of first-order. There is a spontaneous junction magnetization at the bulk critical point.