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arXiv:cond-mat/9803230 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Semi-infinite anisotropic spherical model: Correlations at T>~Tc

Dmitry A. Garanin

Published Jul 1, 1998• 6 citations
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Abstract

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 $D$-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, $d<~2$, at finite temperatures. The correlation functions and the energy density profile in the semi-infinite ASM are calculated analytically and numerically for $T>~{T}_{c}$ and $1<~d<~\ensuremath{\infty}$. Since the lattice dimensionalities $d=1$, 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 $2<d<4$ of Bray and Moore [Phys. Rev. Lett. 38, 735 (1977); J. Phys. A 10, 1927 (1977)].

DOI
10.1103/physreve.58.254

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The ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model (ASM), which is the limit D\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty}D\ensuremath→​\ensuremath∞ 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∞1<~d<~\ensuremath{\infty}1< d< \ensuremath∞. Since the lattice dimensionalities d=1d=1d=1, 2, 3, and 4 are special, a continuous spatial dimensionality d\ensuremath′=d\ensuremath−1{d}^{\ensuremath{'}}=d\ensuremath{-}1d\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 of a semi-infinite nnn-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with me{m}_{e}me​ (easy-magnetization) components take a critical value above which me{m}_{e}me​-component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis (me=1{m}_{e}=1me​=1) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.
The critical behavior at the special surface transition and crossover bevavior from special to ordinary surface transition in semi-infinite n-component anisotropic cubic models are investigated by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation. The crossover behavior for random semi-infinite Ising-like system, which is the nontrivial particular case of the cubic model in the limit n→0n\to 0n→0, is also investigated. The numerical estimates of the resulting two-loop series expansions for the critical exponents of the special surface transition, surface crossover critical exponent ΦΦΦ and the surface critical exponents of the layer, α1α_{1}α1​, and local specific heats, α11α_{11}α11​, are computed by means of Pade and Pade-Borel resummation techniques. For nncnn_{c}nnc​ the cubic fixed point is stable, where ncn_{c}nc​ is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behavior of semi-infinite systems with cubic anisotropy is characterized by new set of surface critical exponents for n>ncn>n_{c}n>nc​.
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.
The critical behaviour of semi-infinite d -dimensional systems with short-range interactions and an O( n ) invariant Hamiltonian is investigated at an m -axial Lifshitz point with an isotropic wave-vector instability in an m -dimensional subspace of d parallel to the surface. Continuum |ϕ| 4 models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant λ) must be included in addition to the familiar ones ∝ϕ 2 . Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m /2 − dimensions (with > 0) are located at λ = λ* = O( ). At second order in , the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order 2 are presented for the surface critical exponent β ord 1 of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by d + m (θ − 1), where θ = ν l 4 /ν l 2 is the bulk anisotropy exponent.
The critical behaviour at the ordinary transition in semi-infinite n -component anisotropic cubic models is investigated by applying the field theoretic approach in d = 3 dimensions up to the two-loop approximation. Numerical estimates of the resulting two-loop series expansions for the critical exponents of the ordinary transition are computed by means of Padé resummation techniques. For n < n c the system belongs to the universality class of the isotropic n -component model, while for n > n c the cubic fixed point becomes stable, where n c < 3 is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behaviour of the semi-infinite systems with cubic anisotropy is characterized by a new set of surface critical exponents for n > n c .
The critical behavior of a system of nnn-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. To first order in $\ensuremath{\epsilon}$ the correlation and susceptibility exponents in the isotropic limit are $2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}$, while the anisotropy or crossover exponent is $\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}$. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ these expansions agree with exact spherical-model results. For n=3n=3n=3 and d=3d=3d=3 series expansions indicate $\ensuremath{\varphi}\ensuremath{\simeq}1.2$ (compared with $\ensuremath{\gamma}\ensuremath{\simeq}1.38$).