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Physical Review Letters

Critical Behavior of a Semi-infinite System: n -Vector Model in the Large- n Limit

Alan J. Bray, M A Moore

Published Apr 4, 1977• 69 citations
Publisher

Abstract

The $n$-component Ginzburg-Landau-Wilson model for a semi-infinite system is solved exactly at $T={T}_{c}$ 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 $d$ in the range $2<d<4$, where $z$, ${z}^{\ensuremath{'}}$ 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.

DOI
10.1103/physrevlett.38.735

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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 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 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 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 ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model (ASM), which is the limit $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty}$ of the DDD-component uniaxially anisotropic classical vector model. The bulk limit of the ASM is similar to that of the spherical model, apart from the role of the anisotropy stabilizing ordering for low lattice dimensions, d< 2d<~2d< 2, at finite temperatures. The correlation functions and the energy density profile in the semi-infinite ASM are calculated analytically and numerically for T> TcT>~{T}_{c}T> Tc​ and $1<~d<~\ensuremath{\infty}$. Since the lattice dimensionalities d=1d=1d=1, 2, 3, and 4 are special, a continuous spatial dimensionality ${d}^{\ensuremath{'}}=d\ensuremath{-}1$ has been introduced for dimensions parallel to the surface. However, preserving a discrete layer structure perpendicular to the surface avoids unphysical surface singularities and allows numerical solitions that reveal significant short-range features near the surface. The results obtained generalize the isotropic-criticality results for 2<d<42<d<42<d<4 of Bray and Moore [Phys. Rev. Lett. 38, 735 (1977); J. Phys. A 10, 1927 (1977)].
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.