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Physical review. B, Condensed matter

Surface critical behavior in the presence of linear or cubic weak surface fields

Hans Werner Diehl, A. Ciach

Published Oct 1, 1991• 30 citations
Publisher

Abstract

The effects of weak symmetry-breaking surface fields on the surface critical behavior of semi-infinite systems with a scalar order parameter are investigated near a bulk critical point. Previous field-theoretic analyses in 4-\ensuremath{\epsilon} dimensions are generalized by incorporating an additional surface term ${\mathit{w}}_{0}$${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ in the Hamiltonian. The mapping of microscopic models of the lattice-gas type on a continuum field theory is reconsidered, and we conclude that the cubic surface field ${\mathit{w}}_{0}$ generically does not vanish both in the case of an Ising ferromagnet in an applied magnetic field H (where ${\mathit{w}}_{0}$\ensuremath{\sim}H as H\ensuremath{\rightarrow}0) as well as in the case of critical adsorption of fluids onto a wall or an interface. Explicit results of a perturbative renormalization-group analysis are given, which show that a fixed point with nonvanishing cubic surface field ${\mathit{w}}^{\mathrm{*}}$=O(\ensuremath{\surd}\ensuremath{\epsilon} ) exists but that at order \ensuremath{\epsilon} the usual ${\mathit{w}}^{\mathrm{*}}$=0 fixed point is stable in the w direction. The associated correction-to-scaling exponent is ${\mathrm{\ensuremath{\omega}}}_{\mathit{w}}$=\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). It is shown that the ${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ term leads to a mixing of surface enhancement and surface field in the leading odd relevant surface scaling field. Furthermore, it is proven beyond perturbation theory that a (redundant) surface operator exists, that corresponds to a similar mixing in the leading even surface scaling field.The implications of this mixing for the critical behavior near the special surface transition are discussed both for the case of Ising ferromagnets as well as that of fluids. In the case of fluids, the mixing entails that quenched surface disorder of the random bond type is sufficient to prevent the occurrence of a (multicritical) special transition for bulk dimensions d\ensuremath{\le}3 A number of other redundant surface operators are identified, and a general redundancy criterion for surface operators (analogous to Wegner's for bulk operators) is given. Finally, a fixed point with ${\mathit{w}}^{\mathrm{*}}$=iO(\ensuremath{\surd}\ensuremath{\epsilon} ) is obtained, which describes the Yang-Lee edge singularities of an equivalent (d-1)-dimensional bulk system with long-range interactions. The associated critical exponents can be determined to all orders in \ensuremath{\epsilon}.

DOI
10.1103/physrevb.44.6642

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Physical review. B, Condensed matter•Jul 1, 1984•40 citations
The effects of weak symmetry-breaking surface fields on the surface critical behavior of semi-infinite systems with a scalar order parameter are investigated near a bulk critical point. Previous field-theoretic analyses in 4-\ensuremath{\epsilon} dimensions are generalized by incorporating an additional surface term {\mathit{w}}_{0}$${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3} in the Hamiltonian. The mapping of microscopic models of the lattice-gas type on a continuum field theory is reconsidered, and we conclude that the cubic surface field w0{\mathit{w}}_{0}w0​ generically does not vanish both in the case of an Ising ferromagnet in an applied magnetic field H (where w0{\mathit{w}}_{0}w0​\ensuremath{\sim}H as H\ensuremath{\rightarrow}0) as well as in the case of critical adsorption of fluids onto a wall or an interface. Explicit results of a perturbative renormalization-group analysis are given, which show that a fixed point with nonvanishing cubic surface field w∗{\mathit{w}}^{\mathrm{*}}w∗=O(\ensuremath{\surd}\ensuremath{\epsilon} ) exists but that at order \ensuremath{\epsilon} the usual w∗{\mathit{w}}^{\mathrm{*}}w∗=0 fixed point is stable in the w direction. The associated correction-to-scaling exponent is \ensuremathωw{\mathrm{\ensuremath{\omega}}}_{\mathit{w}}\ensuremathωw​=\ensuremath{\epsilon}+O(\ensuremathϵ2{\mathrm{\ensuremath{\epsilon}}}^{2}\ensuremathϵ2). It is shown that the \ensuremathφs3{\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}\ensuremathφs3​ term leads to a mixing of surface enhancement and surface field in the leading odd relevant surface scaling field. Furthermore, it is proven beyond perturbation theory that a (redundant) surface operator exists, that corresponds to a similar mixing in the leading even surface scaling field.The implications of this mixing for the critical behavior near the special surface transition are discussed both for the case of Ising ferromagnets as well as that of fluids. In the case of fluids, the mixing entails that quenched surface disorder of the random bond type is sufficient to prevent the occurrence of a (multicritical) special transition for bulk dimensions d\ensuremath{\le}3 A number of other redundant surface operators are identified, and a general redundancy criterion for surface operators (analogous to Wegner's for bulk operators) is given. Finally, a fixed point with w∗{\mathit{w}}^{\mathrm{*}}w∗=iO(\ensuremath{\surd}\ensuremath{\epsilon} ) is obtained, which describes the Yang-Lee edge singularities of an equivalent (d-1)-dimensional bulk system with long-range interactions. The associated critical exponents can be determined to all orders in \ensuremath{\epsilon}.
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We study the surface critical behavior of semi-infinite systems with an n-component order parameter and short-range interactions at a bulk tricritical point. The special surface transition is investigated in detail for space dimensionalities d=3-\ensuremath{\epsilon}\ensuremath{\le}3 using field-theoretic renormalization-group methods. For \ensuremath{\epsilon}>0 its infrared-stable fixed point is described by a Hamiltonian which contains a \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}${\ensuremath{\Vert}}^{4}$ surface interaction of strength O(${\ensuremath{\epsilon}}^{1/2}$) in addition to the familiar \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\rightarrow}${\ensuremath{\Vert}}^{6}$ bulk interaction of strength O(\ensuremath{\epsilon}). For d\ensuremath{\equiv}3, logarithmic corrections of a non-power-law form appear. The renormalization of a proper continuum model is expounded and the implications of the results are discussed, especially with regard to the adsorption of polymers from a \ensuremath{\theta} solution.
We study the special point in the phase diagram of a semi-infinite system where the bulk transition is in the three-dimensional Ising universality class. To this end we performed a finite-size-scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interaction. In order to check for the effect of leading bulk corrections we also simulated the spin-1/2 Ising model on the simple cubic lattice. We accurately estimated the surface enhancement coupling at the special point of these models. We find yt1=0.718(2){y}_{{t}_{1}}=0.718(2)yt1​​=0.718(2) and yh1=1.6465(6){y}_{{h}_{1}}=1.6465(6)yh1​​=1.6465(6) for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.
We study the surface critical behavior of semi-infinite systems belonging to the bulk universality class of the Ising model. Special attention is paid to the local behavior of experimentally relevant quantities such as the order parameter and the correlation function in the crossover regimes between different surface universality classes, where the surface field h1 and the temperature enhancement c can induce additional macroscopic length scales. Starting from the field-theoretical φ4 model and employing renormalization-group improved perturbation theory (ϵ-expansion), explicit results for the local behavior of the correlation and structure function (zero-loop) and the order parameter (one-loop) are derived. Supplementing earlier studies that focused on the special transition, we here pay particular attention to the situation where a large c suppresses the tendency to order in the surface (ordinary transition) but a surface field h1 generates a small surface magnetization m1. Our results are in good agreement with recent phenomenological considerations and Monte Carlo studies devoted to similar questions and, combined with the latter, provide a much more detailed understanding of the local properties of systems with weak surface fields.
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 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 $\ensuremath{\epsilon}$.