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Journal of Physics Condensed Matter

The critical wetting saga: how to draw the correct conclusion

Andrew O. Parry, C. Rascón, N. R. Bernardino, José Enríque

Published Nov 12, 2008• 11 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/20/49/494234

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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.
We consider two-dimensional (d = 2) systems with short-ranged microscopic interactions, where interface unbinding (wetting) transitions occur in the limit of vanishing temperature T .For T = 0 the transition is characterized by non-universal critical properties analogous to those established for thermal wetting transitions in d = 3, albeit with a redefined capillary parameter ω.Within a functional renormalization-group treatment of an effective interfacial model, we compute the finite-temperature phase diagram, exhibiting a line of interface unbinding transitions, terminating at T = 0 with an interfacial quantum-critical point.We identify distinct scaling regimes, reflecting the interplay between quantum and thermal interfacial fluctuations.A crossover line marking the onset of the quantum-critical regime is described by the d = 3 interfacial correlationlength exponent ν || .This potentially opens another way to investigate the non-universal character of ν || .On the other hand, the emergent interfacial quantum-critical regime shows no signatures of non-universality.
We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields [E. V. Albano and K. Binder, Phys. Rev. Lett. 109, 036101 (2012)] establishes that the average magnetization of the sample, with critical exponent $\ensuremath{\beta}$, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by $\ensuremath{\gamma}+2\ensuremath{\beta}={\ensuremath{\nu}}_{\ensuremath{\parallel}}+{\ensuremath{\nu}}_{\ensuremath{\perp}}$ requires $\ensuremath{\beta}=1/2$ ($\ensuremath{\gamma}=4, {\ensuremath{\nu}}_{\ensuremath{\parallel}}=3$, and ${\ensuremath{\nu}}_{\ensuremath{\perp}}=2$), the thermodynamic scaling establishes that ${\mathrm{\ensuremath{\Delta}}}_{s}=\ensuremath{\gamma}+\ensuremath{\beta}$, which in contrast requires $\ensuremath{\beta}=0$ (${\mathrm{\ensuremath{\Delta}}}_{s}=4$), where $\ensuremath{\gamma}, {\ensuremath{\nu}}_{\ensuremath{\parallel}}, {\ensuremath{\nu}}_{\ensuremath{\perp}}$, and ${\mathrm{\ensuremath{\Delta}}}_{s}$ are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., $\ensuremath{\beta}=0$) and a direct measurement of the susceptibility critical exponent $\ensuremath{\gamma}/{\ensuremath{\nu}}_{\ensuremath{\perp}}=2.0\ifmmode\pm\else\textpm\fi{}0.2$, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise.
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.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
We study how the formalism of the hierarchical reference theory (HRT) can be extended to inhomogeneous systems. HRT is a liquid-state theory which implements the basic ideas of the Wilson momentum-shell renormalization group (RG) to microscopic Hamiltonians. In the case of homogeneous systems, HRT provides accurate results even in the critical region, where it reproduces scaling and nonclassical critical exponents. We applied the HRT to study wetting critical phenomena in a planar geometry. Our formalism avoids the explicit definition of effective surface Hamiltonians but leads, close to the wetting transition, to the same renormalization group equation already studied by RG techiques. However, HRT also provides information on the nonuniversal quantities because it does not require any preliminary coarse graining procedure. A simple approximation to the infinite HRT set of equations is discussed. The HRT evolution equation for the surface free energy is numerically integrated in a semi-infinite three-dimensional Ising model and the complete wetting phase transition is analyzed. A renormalization of the adsorption critical amplitude and of the wetting parameter is observed. Our results are compared to available Monte Carlo simulations.