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Europhysics Letters (EPL)

Susceptibility Critical Amplitude Ratios and Tests of the Stiffness Instability at Wetting Transitions

Andrew O. Parry, C. J. Boulter

Published Nov 1, 1994• 3 citations
Publisher

Abstract

We carefully consider the critical amplitude ratios which may be defined for the local susceptibility at critical wetting transitions. In the strong-fluctuation regime scaling ideas suggest that there are two universal amplitude ratios (denoted R and R z ) which reflect the criticality a ) exactly at the wall and b ) related to the short-distance expansion. R and R z are calculated explicitly in dimension d = 2 and for different mean-field theories where we find that the amplitude relation R = R z is obeyed. In d = 3, corresponding to the marginal dimensionality, R is calculated using a linear renormalization group approach for a rather general model yielding non-universal behaviour. Here r is shown to satisfy a simple amplitude-exponent relation. We argue that future simulation studies of the amplitude ratio for a novel modified Ising model should provide a stringent test of the stiffness-instability theory of Fisher and Jin.

DOI
10.1209/0295-5075/28/4/006

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We carefully consider the critical amplitude ratios which may be defined for the local susceptibility at critical wetting transitions. In the strong-fluctuation regime scaling ideas suggest that there are two universal amplitude ratios (denoted R and R z ) which reflect the criticality a ) exactly at the wall and b ) related to the short-distance expansion. R and R z are calculated explicitly in dimension d = 2 and for different mean-field theories where we find that the amplitude relation R = R z is obeyed. In d = 3, corresponding to the marginal dimensionality, R is calculated using a linear renormalization group approach for a rather general model yielding non-universal behaviour. Here r is shown to satisfy a simple amplitude-exponent relation. We argue that future simulation studies of the amplitude ratio for a novel modified Ising model should provide a stringent test of the stiffness-instability theory of Fisher and Jin.
A long-standing problem in condensed matter physics concerns the nature of the critical wetting phase transition in the Ising model or, more generally, in 3D systems with short-ranged forces.This is of fundamental interest because 3D corresponds to the upper critical dimension of the transition and it is not clear a priori whether the behaviour of the system will be mean-field-like or fluctuation-dominated.Renormalization group studies of the standard coarse-grained effective interfacial Hamiltonian model famously predict strong non-universal critical exponents which depend on the value of the so-called wetting parameter ω.However, these predictions are at odds with extensive Monte Carlo simulations of wetting in the Ising model, due to Binder, Landau and coworkers, which appear to be more mean-field-like.Further amendments to the interfacial Hamiltonian, which included the presence of a position-dependent stiffness, worsened the problem by paradoxically predicting fluctuation-induced first-order wetting behaviour.Here we show from re-analysis of a microscopic Landau-Ginzburg-Wilson model of 3D short-ranged wetting that correlation functions are characterized by two diverging parallel length scales, not one, as previously thought.This has a simple diagrammatic explanation using a non-local interfacial Hamiltonian and yields a thermodynamically consistent theory of wetting in keeping with exact sum rules.The non-local model crucially contains long-ranged two-body interfacial interactions, characterized by the new length scale, which were missing in earlier treatments.For critical wetting the second length cuts off the spectrum of interfacial fluctuations determining the repulsion from the wall.We show how this corrects previous renormalization group predictions for fluctuation effects, based on local interfacial Hamiltonians.In particular, lowering the cut-off leads to a substantial reduction in the effective value of the wetting parameter controlling the non-universality and also prevents the transition being driven first-order.Quantitative comparison with the Ising model simulation studies is also made.
The author considers the finite-size (FS) scaling behaviour of the transverse correlation lengths xi /sub /// (n) for an interface confined in a strip geometry exactly at a second-order fluctuation-dominated wetting transition. He shows that the dimensionless amplitude ratios A/sub /// (n) = Sigma L 2 /k B T xi /sub /// (n) (with Sigma the surface stiffness coefficient) are related to the short-distance expansion critical exponent theta by A/sub /// (n) = pi 2 n(n+ theta -1)/2 for all fluctuation-dominated wetting transitions, provided the binding potential V(y) in the capillary-wave model is conformally mapped from the semi-infinite plane. He argues that the existence of this amplitude-exponent relation reflects the conformal invariance of the capillary-wave model Lagrange density and lists a number of one-point functions which can be shown to exhibit this invariance explicitly.
The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close to their critical point. Values of the universal ratios involving correlation length and specific-heat amplitudes are obtained from the $\ensuremath{\epsilon}$ expansion, for Ising, $X\ensuremath{-}Y$, and Heisenberg models. In the latter two cases the correlation function has a power-law behavior at large distances below Tc{T}_{c}Tc​, and the (transverse) correlation length is defined in terms of the stiffness constant ${\ensuremath{\rho}}_{s}$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid 4He^{4}\mathrm{He}4He, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet RbMnF3{\mathrm{RbMnF}}_{3}RbMnF3​. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.
We propose a new class of universal amplitude ratios which involve the first terms of the short distance expansion of the correlators of a statistical model in the vicinity of a critical point. We will describe the critical system with a conformal field theory (UV fixed point) perturbed by an appropriate relevant operator. In two dimensions the exact knowledge of the UV fixed point allows for accurate predictions of the ratios and in many nontrivial integrable perturbations they can even be evaluated exactly. In three-dimensional O ( N ) scalar systems, feasible extensions of some existing results should allow perturbative expansions for the ratios to be obtained. By construction these universal ratios are a perfect tool to explore the short distance properties of the underlying quantum field theory even in regimes where the correlation length and one-point functions are not accessible in experiments or simulations.
Abstract We discuss how to calculate non-equilibrium universal amplitude ratios in the functional renormalization group approach, extending its applicability. In particular, we focus on the critical relaxation of the Ising model with non-conserved dynamics (model A) and calculate the universal amplitude ratio associated with the fluctuation–dissipation ratio of the order parameter, considering a critical quench from a high-temperature initial condition. Our predictions turn out to be in good agreement with previous perturbative renormalization-group calculations and Monte Carlo simulations.