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Journal of Physics A Mathematical and General

Anomalous decay of pair correlations for two-dimensional critical wetting

Andrew O. Parry

Published Apr 21, 1992• 4 citations
Publisher

Abstract

Develops a scaling theory for the decay of the spin-spin correlation function at two-dimensional (d=2) wetting transitions in systems with short-ranged forces. For the critical wetting transition effective Hamiltonian results show that the scaling theory is obeyed but the decay of correlations is anomalous. In contrast, for the complete wetting transition, scaling and the Ornstein-Zernike theory are both valid. The author argues that the anomalous decay is specific to zero bulk field critical wetting transitions ad d=2.

DOI
10.1088/0305-4470/25/8/020

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Physical Review Letters•Mar 8, 1971•45 citations
Develops a scaling theory for the decay of the spin-spin correlation function at two-dimensional (d=2) wetting transitions in systems with short-ranged forces. For the critical wetting transition effective Hamiltonian results show that the scaling theory is obeyed but the decay of correlations is anomalous. In contrast, for the complete wetting transition, scaling and the Ornstein-Zernike theory are both valid. The author argues that the anomalous decay is specific to zero bulk field critical wetting transitions ad d=2.
Clarification of critical wetting with short-range forces by simulations has been hampered by the lack of accurate methods to locate where the transition occurs. We solve this problem by developing an anisotropic finite-size scaling approach and show that then the wetting transition is a "bulk" critical phenomenon with order parameter exponent equal to zero. For the Ising model in two dimensions, known exact results are straightforwardly reproduced. In three dimensions, it is shown that previous estimates for the location of the transition need revision, but the conclusions about a slow crossover away from mean-field behavior remain unaltered.
In this work the decay of correlation in the ddd-dimensional Ising model is studied at low temperatures as a function of dimensionality of the lattice and magnetic field hhh. Except for the special case of the two-dimensional zero-field nearest-neighbor lattices, the decay of correlation verifies the Ornstein-Zernike prediction ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}(d,h){R}^{\ensuremath{-}\frac{(d\ensuremath{-}1)}{2}}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. For the two-dimensional zero-field case, the Ornstein-Zernike form is replaced by the "anomalous" form ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}{R}^{\ensuremath{-}2}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. This "anomalous" result is shown to arise from the peculiarities of the spectrum of the transfer matrix in this case and is replaced by the Ornstein-Zernike result when further-neighbor forces are present. The results presented herein agree with the previously obtained exact results for the zero-field two-dimensional Ising model.
The global phase diagram of wetting in the two-dimensional Ising model is obtained through the exact calculation of the surface excess free energy. In addition to a surface field for inducing wetting, a surface-coupling enhancement is also included. The wetting transition (of second order) is critical for any finite ratio of surface coupling J_{s} to bulk coupling J, and becomes of first order in the limit J_{s}/J→∞. However, for J_{s}/J≫1, the critical region is exponentially small and is practically invisible to numerical studies. A distinct preasymptotic regime exists in which the transition displays first-order character. In this regime, surprisingly, the surface susceptibility and surface specific heat develop a divergence and show anomalous scaling with an exponent equal to 3/2.
Abstract. Effective Hamiltonian models predict non-universal critical singularities for two--dimensional wetting transitions with long-ranged forces. We verify these predictions by studying delocalization transitions in an infinitely long Ising strip, of width L (lattice spacings), with long-ranged surface fields that have opposite sign at each surface. The extrapolated asymptotic value for the exponent βs does not confirm to the predicted non-universality but instead approaches the same universal value representative of systems with short-ranged forces. The crossover of the scaling behaviour of the transition lines is presented. Moreover, contrary to the existing predictions, the critical wetting transition for p = 2 has been found.
Effective Hamiltonian models predict nonuniversal critical singularities for two-dimensional wetting transitions with marginal long-ranged forces. We test these predictions by studying interfacial delocalization transitions in an infinitely long Ising strip, of width L (lattice spacings), with external fields that are long ranged and have opposite signs at each surface. Finite-size scaling suggests that the shift of the delocalization temperature T(c)(L) below the (semi-infinite) wetting temperature T(w) scales as L(-1/beta(s)) with beta(s) the adsorption critical exponent. Density-matrix renormalization-group methods allow us to study the behavior of T(c)(L) for L up to several hundred lattice spacings. For short-ranged forces the method recovers the universal value of beta(s)=1 known from the exact solution. While marginal long-ranged forces strongly influence the finite-size scaling of T(c)(L) , the extrapolated asymptotic value for the exponent beta(s) does not appear to confirm the predicted nonuniversality, but instead approaches the same universal value representative of systems with short-ranged forces.
The asymptotic decay of the two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ at and near a phase-separation point is discussed for ddd-dimensional, spin-\textonehalf{} Ising models at low temperature. The general behavior, even on the phase boundary (H=0H=0H=0), is in agreement with extended Ornstein-Zernike predictions. It is shown why the nearest-neighbor two-dimensional model in zero field is an exception. The decay of ${G}_{\mathrm{AB}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{1},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{2})$ near a free surface at high temperatures agrees with phenomenological predictions using a vanishing boundary condition. At low temperature, however, the decay of correlation near a surface is exponentially slower than in the bulk.