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Physical review. B, Solid state

Critical phenomena in semi-infinite systems. I. ε expansion for positive extrapolation length

T. C. Lubensky, Morton H. Rubin

Published Jun 1, 1975• 127 citations
Publisher

Abstract

The Wilson-Fisher $\ensuremath{\epsilon}$ expansion is used to calculate critical exponents to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for $n$-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.

DOI
10.1103/physrevb.11.4533

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The Wilson-Fisher \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion is used to calculate critical exponents to first order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, all surface exponents can be calculated from bulk exponents and a single surface exponent, \ensuremathη\ifmmode~\else˜\fi=(12)\ensuremathϵ(n+2)(n+8)\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\frac{(\frac{1}{2})\ensuremath{\epsilon}(n+2)}{(n+8)}\ensuremathη\ifmmode~\else˜\fi​=(n+8)(21​)\ensuremathϵ(n+2)​, describing the rate at which bulk correlation functions are approached when all coordinates are far from the surface. The exponents \ensuremathη\ensuremath⊥{\ensuremath{\eta}}_{\ensuremath{\perp}}\ensuremathη\ensuremath⊥​ and \ensuremathη\ensuremath∥{\ensuremath{\eta}}_{\ensuremath{\parallel}}\ensuremathη\ensuremath∥​ introduced by Binder and Hohenberg are, respectively, 1\ensuremath−\ensuremathη\ifmmode~\else˜\fi1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}1\ensuremath−\ensuremathη\ifmmode~\else˜\fi​ and 2(1\ensuremath−\ensuremathη\ifmmode~\else˜\fi)2(1\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}})2(1\ensuremath−\ensuremathη\ifmmode~\else˜\fi​). A form for the fixed-point spin correlation valid for all dimensions containing only the exponents \ensuremathη\ensuremath{\eta}\ensuremathη and \ensuremathη\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremathη\ifmmode~\else˜\fi​ is proposed. With this form, all critical exponents for a semi-infinite system can be obtained from \ensuremathη\ensuremath{\eta}\ensuremathη, \ensuremathν\ensuremath{\nu}\ensuremathν, and \ensuremathη\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremathη\ifmmode~\else˜\fi​ if scaling is assumed.
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.
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 .
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 nnn-component Ginzburg-Landau-Wilson model for a semi-infinite system is solved exactly at T=TcT={T}_{c}T=Tc​ in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In the scaling regime the spin-spin correlation function is $G(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}, z, {z}^{\ensuremath{'}}, {T}_{c})=\mathrm{const}{{{[{\ensuremath{\rho}}^{2}+{(z\ensuremath{-}{z}^{\ensuremath{'}})}^{2}]}^{\ensuremath{-}1}\ensuremath{-}{[{\ensuremath{\rho}}^{2}+{(z+{z}^{\ensuremath{'}})}^{2}]}^{\ensuremath{-}1}}}^{\frac{(d\ensuremath{-}2)}{2}}$, for dimensionalities ddd in the range 2<d<42<d<42<d<4, where zzz, ${z}^{\ensuremath{'}}$ are the distances of the two spins from the surface and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ is their separation parallel to the surface. The critical exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ are $\frac{(d\ensuremath{-}2)}{2}$ and ($d\ensuremath{-}2$), respectively.
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.