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arXiv:cond-mat/9409064 · Physical Review Letters

Surface Critical Behavior in Fixed Dimensions d<4 : Nonanalyticity of Critical Surface Enhancement and Massive Field Theory Approach

Hans Werner Diehl, Mykola A. Shpot

Published Dec 19, 1994• 52 citations
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Abstract

The critical behavior of semi-infinite systems in fixed dimensions $d<4$ is investigated theoretically. The appropriate extension of Parisi's massive field theory approach is presented. Two-loop calculations and subsequent Pad\'e-Borel analyses of surface critical exponents of the special and ordinary phase transitions yield estimates in reasonable agreement with recent Monte Carlo results. This includes the crossover exponent $\ensuremath{\Phi}(d=3)$, for which we obtain the values $\ensuremath{\Phi}(n=1)\ensuremath{\simeq}0.54$ and $\ensuremath{\Phi}(n=0)\ensuremath{\simeq}0.52$, considerably lower than the previous $\ensuremath{\epsilon}$-expansion estimates.

DOI
10.1103/physrevlett.73.3431

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Physical review. B, Condensed matter•Oct 1, 1991•30 citations
The critical behavior of semi-infinite systems in fixed dimensions d<4d<4d<4 is investigated theoretically. The appropriate extension of Parisi's massive field theory approach is presented. Two-loop calculations and subsequent Pad'e-Borel analyses of surface critical exponents of the special and ordinary phase transitions yield estimates in reasonable agreement with recent Monte Carlo results. This includes the crossover exponent \ensuremathΦ(d=3)\ensuremath{\Phi}(d=3)\ensuremathΦ(d=3), for which we obtain the values \ensuremathΦ(n=1)\ensuremath≃0.54\ensuremath{\Phi}(n=1)\ensuremath{\simeq}0.54\ensuremathΦ(n=1)\ensuremath≃0.54 and \ensuremathΦ(n=0)\ensuremath≃0.52\ensuremath{\Phi}(n=0)\ensuremath{\simeq}0.52\ensuremathΦ(n=0)\ensuremath≃0.52, considerably lower than the previous \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-expansion estimates.
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 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​.
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.
We use a three-dimensional massive field theory up to the two-loop approximation to study the critical behavior of semi-infinite quenched random Ising-like systems at the special surface transition. Besides, we extend up to the next-to leading order, the previous first-order results of the sqrt[epsilon] expansion obtained by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. The numerical estimates for surface critical exponents in both cases are computed by means of the Padé analysis. Moreover, in the case of the massive field theory we perform Padé-Borel resummation of the resulting two-loop series expansions for surface critical exponents. The most reliable estimates for critical exponents of semi-infinite systems with quenched bulk randomness at the special surface transition, which we can obtain in the frames of the present approximation scheme, are eta(//)=-0.238, Delta(1)=1.098, eta( perpendicular )=-0.104, beta(1)=0.258, gamma(11)=0.839, gamma(1)=1.426, delta(1)=6.521, and delta(11)=4.249. These values are different from critical exponents for pure semi-infinite Ising-like systems and show that in a system with quenched bulk randomness the plane boundary is characterized by a new set of critical exponents at the special surface transition.
The critical behavior of a semi-infinite nnn-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions near the special transition is studied. The renormalization of the relevant surface operators is discussed and explicitly carried out in two-loop order. It is found that the crossover exponent ${\ensuremath{\varphi}}_{s}$ related to the square of the order parameter in the surface does not satisfy the relation ${\ensuremath{\varphi}}_{s}=1\ensuremath{-}\ensuremath{\nu}$ due to Bray and Moore.
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 w0{\mathit{w}}_{0}w0​${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ 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 ${\mathrm{\ensuremath{\omega}}}_{\mathit{w}}$=\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 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}.