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Journal of Physics A Mathematical and General

Semi-infinite ϕ4systems: ε expansions for susceptibility exponents

Jeff S. Reeve, Anthony John Guttmann

Published Mar 1, 1981• 0 citations
Publisher

Abstract

The surface and local susceptibility exponents gamma 1 and gamma 11 for semi-infinite phi 4 , O(n) systems are calculated to order epsilon =4-d at the ordinary bulk transition temperature. The results agree with the predictions of correlation function scaling.
DOI
10.1088/0305-4470/14/3/016

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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 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.
The critical behavior of a continuous spin system in a semi-infinite sample is studied for all values of the extrapolation length $\ensuremath{\lambda}$ using mean-field theory. A new transition, which we have called the extraordinary transition, is found for $\ensuremath{\lambda}<0$ in which the bulk orders at a temperature below the surface ordering temperature. In this paper we have calculated the magnetic susceptibilities $\ensuremath{\chi}(z)$ and $\ensuremath{\chi}(z, z)$ and the correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}})$ at all the phase transitions. We have used these results to compute the various $\ensuremath{\gamma}$ and $\ensuremath{\eta}$ exponents and to study the scaling relations introduced by Binder and Hohenberg.
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 .