Renormalization group calculations for wetting transitions of infinite order and continuously varying order. I. Local interface Hamiltonian approach.

# Renormalization group calculations for wetting transitions of infinite order and continuously varying order. I. Local interface Hamiltonian approach.

J.O. Indekeu, K. Koga, H. Hooyberghs and A.O. Parry Institute for Theoretical Physics, KU Leuven, BE-3001 Leuven, Belgium
Department of Chemistry, Faculty of Science, Okayama University, Okayama 700-8530, Japan
Department of Mathematics, Imperial College London, London SW7 2BZ, UK
July 15, 2019
###### Abstract

We study the effect of thermal fluctuations on the wetting phase transitions of infinite order and of continuously varying order, recently discovered within a mean-field density-functional model for three-phase equilibria in systems with short-range forces and a two-component order parameter. Using linear functional renormalization group (RG) calculations within a local interface Hamiltonian approach, we show that the infinite-order transitions are robust. The exponential singularity (implying ) of the surface free energy excess at infinite-order wetting as well as the precise algebraic divergence (with ) of the wetting layer thickness are not modified as long as , with the dimensionless wetting parameter that measures the strength of thermal fluctuations. The interface width diverges algebraically and universally (with ). In contrast, the non-universal critical wetting transitions of finite but continuously varying order are modified when thermal fluctuations are taken into account, in line with predictions from earlier calculations on similar models displaying weak, intermediate and strong fluctuation regimes.

preprint: APS/123-QED

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## I Introduction

In recent work Letter (); Long () wetting transitions of infinite order were uncovered in a mean-field density functional theory (DFT) for systems with short-range forces and with a two-component order parameter. Although infinite-order wetting transitions were known to show up in certain fluctuation regimes studied using functional renormalization group (RG) theory for wetting or related methods FH (); LN (); BHL (); LKZ (); KL2 (); KL1 (), it was a surprise that they can appear prominently already at mean-field level. A revisitation of a variety of early DFT’s for wetting H (); W () has led to the conclusion Long () that segments of infinite-order wetting transitions must be fairly ubiquitous, but have apparently long been overlooked, in models with a multi-component order parameter. These segments typically connect a regime of first-order wetting to one of critical wetting with continuously varying (non-universal) critical exponents.

We start with recalling briefly the main ingredients of the model, which has been described and discussed in detail in two earlier papers Letter (); Long (). The mean-field DFT is defined through a functional of two spatially varying densities, or density components, and . It represents the excess free energy per unit area of an interface, oriented perpendicular to ,

 ^σ[ρ1,ρ2]=∫∞−∞dz⟨{12(∇ρ1(r))2+12(∇ρ2(r))2+F(ρ1(r),ρ2(r);a,b)}⟩{x,y}, (1)

The outer brackets denote that the integrand is averaged over the directions and parallel to the interface. The free-energy per unit volume is the following 6-th order polynomial

 F(ρ1,ρ2;a,b)=((ρ1+1)2+ρ22)((ρ1/a)2+(ρ2−b)2)((ρ1−1)2+ρ22). (2)

The model parameter is an asymmetry variable, being its symmetric value. This presence of may have various grounds. In systems with an obvious geometrical symmetry relating the two densities and , may be related to spatial anisotropy. This is the case, e.g., of a ferromagnet with cubic anisotropy W () or in general of systems that can be mapped onto a magnetic model with a magnetization vector order parameter H (). On the other hand, if the two densities are unrelated by any symmetry, it is always possible to redefine and scale them so that the gradient-squared part of the functional is symmetric (and diagonal) in and , as it is in (1). Without loss of generality, our asymmetry parameter is defined adopting this convention. Different conventions may lead to different definitions for and consequently to differences in -dependent calculational results, but do not affect the ultimate physical results (for the critical exponents, etc.). A model for which in this regard different conventions are used in different works, is, e.g., the Ginzburg-Landau DFT for superconductivity IvLLetter (); IvL (); Long (); vLH (), in which one order parameter pertains to the superconducting wave function and the other to the magnetic vector potential. The model parameter is a control variable that allows wetting to be induced. Although it has the same dimension as the density , it is physically a field-like variable which may depend on temperature and/or other external fields.

For arbitrary and , reaches its minimum value, , when the densities take their bulk-phase values at . These are

 αphase : ρ1=−1,ρ2=0; βphase : ρ1=0,ρ2=b; (3) γphase : ρ1=1,ρ2=0.

As in the pertinent foregoing work Letter (); Long () we study the wetting or nonwetting of the interface by the phase. In the DFT defined through (1) the wetting transition was found to be of first order for . A second-order wetting transition was found for (symmetric model), when is lowered towards , already in KogaWidom2008 (). For that transition the critical exponent associated with the free-energy singularity, , takes the value 2, and the critical exponent of the wetting layer thickness takes the value log, signifying a logarithmic divergence at wetting. Further, a non-universal critical wetting transition was obtained for , upon lowering towards . For this transition, and log. Finally, a segment of infinite-order wetting transitions was found at and for . The singularity in the spreading coefficient near wetting was conjectured to be of the form, in the limit ,

 −S∝e−C/(1−a);withC>0, (4)

implying , and the wetting layer thickness was conjectured to diverge in the algebraic manner, for ,

 ℓ∝(1−a)−1, (5)

implying . These conjectures were based on the analytical solution of a different but related model, for which the leading terms, for large , of an interface potential could be calculated. Furthermore, accurate numerical computations for the original model support the validity of the results (4) and (5). The global wetting phase diagram in the -plane has been presented in Letter (); Long () and is, for the sake of clarity, reproduced in Fig.1 in annotated version.

The remainder of this paper is organized as follows. In Section II we discuss the occurrence of infinite-order wetting transitions in mean-field theories and RG theories. Section III presents a new derivation of an interface potential using the simplest possible crossing criterion for the interface position. Section IV is devoted to RG calculations for wetting transitions that are of infinite order already at mean-field level. In Section V we point out that the renormalization of the wetting transitions of finite but continuously varying order qualitatively reproduces previously obtained RG corrections for similar wetting transitions with, however, interesting quantitative differences. Conclusions are drawn in Section VI.

## Ii Occurrences of infinite-order wetting

### ii.1 Mean-field theories

One of the main conclusions of Long () is that every mean-field DFT whose properties in the vicinity of the phase can be “mapped” onto those of (1) is susceptible of displaying a segment of infinite-order wetting along the “symmetric” line , spanning some range of field variables. However, it must be checked case by case whether the line belongs to the physical subspace in which wetting transitions are possible. Let us comment in this regard on the three examples discussed in Long ().

In the pioneering two-component order parameter DFT of Hauge H () for wetting at a wall the asymmetry parameter is the ratio of two curvatures, and , that characterize the shape of the potential in the vicinity of the bulk wetting-phase point in the plane of the order parameters . For the wetting transition is of first-order. For it is a standard second-order transition (), for it is of higher order and non-universal (), and for is (again) a standard second-order transition (). However, recent insights Long () imply that precisely at the regimes of first- and higher-order wetting are connected by a segment of infinite-order transitions. We infer that this segment extends, in the notation of H (), from to , where is the control parameter that allows one to induce wetting by varying the temperature, through , and/or by varying the surface field . Here, , with and parameters associated with the “free” interface between the wetting and non-wetting phases in bulk and a surface enhancement parameter. We note that in this model the interface potential features the following leading two terms, for (critical wetting)

 V(ℓ)=−τα1e−λ1ℓ−H2α2e−λ2ℓ+O(e−2λ1ℓ), (6)

with a suitably scaled wetting layer thickness.

In the two-component order parameter DFT of Walden et al. W () for investigating the surface properties of a ferromagnet with cubic anisotropy and vector order parameter , a similar scenario unwinds. The asymmetry parameter is again the ratio of two curvatures, that characterize the shape of the potential in the vicinity of the bulk wetting-phase point in the plane of the order parameter components. Here, is the usual reduced temperature distance to the Curie point, . This ratio equals the ratio of two lengths, , with a measure of the magnetic anisotropy ( being the isotropic value). Note that the symmetric value for , , corresponds, however, to , which is already well within the range of values of of materials with “cubic anisotropy”, , to which our attention is restricted from now on. The anisotropy is (strongly) temperature dependent Asselin (). For the wetting transition is of first order. For it is of second order (), for it is of higher order and non-universal (), and for it is universal and of second order (). Also for this model, the revisitation in Long () concluded that precisely at the regimes of first- and higher-order wetting must be connected by a segment of infinite-order transitions. We infer that this segment extends, in the notation of W (), from to , where is the control parameter that allows one to induce wetting by varying the temperature, through , and/or by varying the surface field . Here, is a ratio of parameters associated with the “free” interface between the wetting and non-wetting phases in bulk, is a surface enhancement parameter and is a (constant) length. In this model the interface potential features the following leading two terms, for (critical wetting)

 V(ℓ)=−τξ0a0xe−ℓ/ξx+ξ0Hsya0ye−ℓ/ξy+O(e−2ℓ/ξx). (7)

In the Ginzburg-Landau (GL) theory for superconductivity IvLLetter (); IvL (); vLH (); CI (), after suitable scaling of the two order parameters so as to arrive at a symmetric gradient-squared part of the functional, the asymmetry parameter takes the form , with the GL parameter, being the ratio of the superconducting coherence length to the magnetic penetration depth. Close to the bulk critical point is a material constant, with applicable to type-I superconductors, for which wetting by a macroscopic superconducting layer (Meissner phase), intruding between the normal phase and the sample surface, can be investigated. For materials with first-order wetting is possible IvLLetter (); IvL (), while for non-universal critical wetting is possible IvLLetter (); IvL (); vLH () with . At first sight it would seem that the global wetting phase diagram for this system might feature a segment of infinite-order transitions at , but this is not the case. In the limit the surface tension between the superconducting and normal phases vanishes (bulk multicritical point) and the notion of wetting phase transition ceases to exist. Nevertheless, phenomena reminiscent of enhanced adsorption do occur in this limit CI (); Dietrich (); Speth (). This DFT thus provides an interesting exception to the scenario of infinite-order wetting in DFT’s with two order parameters. For completeness, we also give the leading structure of the interface potential, for ,

 V(ℓ)=−Ae−κ√2ℓ+Be−ℓ+O(e−2κ√2ℓ), (8)

with and given explicitly in vLH ().

The occurrence of infinite-order wetting in DFT’s with continuously varying critical exponents compellingly raises the question whether infinite-order wetting occurs also in the famous van der Waals theory of wetting for short-range forces with a one-component order parameter AH (), featuring a non-universal that depends on the ratio of the inverse bulk correlation length to the inverse range of the exponentially decaying wall-fluid potential. In this model the range of the exponentially decaying effective fluid-fluid potential is set to unity, and the bulk correlation length varies monotonically from unity at to infinity at the bulk critical point at . Consequently, at finite temperatures, . For , the second-order wetting transition of the Sullivan model is recovered with and . For there are several possibilities. Consider first the case . For the wetting transition, if it exists, turns out to be of first order, while for it is critical and non-universal with . The order of the transition may thus be high but cannot diverge since and . Next, for (for arbitrary ) the transition, if it is critical, settles on the usual universal second-order transition. This model therefore constitutes another “counterexample” for which the would-be infinite-order transition (mathematically conceivable at ) does not take place since its location would fall outside the domain of validity of continuous wetting nonthermo (). Note that the border point of this domain lies at , for which only second-order wetting is possible, regardless of the value of . We close this case by recalling that the interface potential for this model takes the form, for large ,

 V(ℓ)=Kλe−λℓ+Kβe−βℓ+O(e−2λℓ), (9)

with analytic expressions for and given in AH ().

### ii.2 Renormalization Group theories and exact results

The first infinite-order wetting transition that enjoyed some attention was that associated with the strong thermal fluctuation regime, , of the short-range critical wetting (SRCW) transition in three dimensions LKZ (); BHL (); FH (). Here, is the “wetting parameter” given by

 ω=kT4πσξ2, (10)

with the interfacial tension between the wetting phase and the bulk phase far from the “wall”, and the bulk correlation length in the wetting phase. For the Ising model near , the universal value applies PREL (). The pioneering functional RG treatments or related variational approaches of SRCW in BHL (); LKZ () already revealed that the transition turns infinite order for , and, complemented with subsequent refined calculations FH (), provided the famous non-universal critical exponents in the weak () and intermediate () fluctuation regimes relevant to the Ising model universality class. Note that the results of the mean-field approximation are formally retrieved in the limit . Interestingly, the global phase diagram of SRCW in (see Fig.2 in FH ()) bears some resemblance to that of the mean-field DFT we discuss, in that a regime of transitions of continuously varying order culminates in an infinite-order transition when diverges, for in the mean-field DFT and for in the functional RG theory. The algebraic divergence of the wetting layer thickness at wetting (with ) is also shared by the MF model and the RG theory. However, the similarity is not complete. In the mean-field DFT the segment of infinite-order transitions is associated with a jump, at , in the control parameter, say temperature and/or surface field, with which wetting can be induced, while in the RG treatment the control parameter at wetting, say , varies continuously (linearly) as a function of for .

In two dimensions an interesting segment of infinite-order wetting transitions has been uncovered in the so-called intermediate fluctuation regime LN (), by means of Feynman path integral and transfer matrix methods, following pioneering analytical work which already established an infinite-order transition KL2 (). In the fluctuation-induced repulsion between an unbinding interface and a wall, is, unlike in , not exponentially but algebraically decaying, in the manner . When this entropic repulsion competes with an attractive direct interaction behaving for large as , with , which is appropriate for certain systems with long-range forces, a wetting transition of infinite order is possible, with and (essential singularities for the spreading coefficient as well as for the wetting layer thickness). The global wetting phase diagram (see Fig.1 in LN ()) displays a segment of infinite-order wetting at , spanning an (infinite) jump in the amplitude of the short-range direct interaction between interface and wall.

## Iii Derivation of an interface potential within mean-field theory

Let us recall the starting point of the interface potential approach adopted in Long (). Expanding the surface free-energy functional (2) about the bulk -phase point in the -plane leads to the approximation

 F(ρ1,ρ2;a,b)≈F(2)(ρ1,ρ2;a,b)≡(1+b2)2[(ρ1/a)2+(ρ2−b)2]. (11)

Solving the Euler-Lagrange equations Long () within this harmonic approximation leads to, with , where ,

 ρ1(Z) = a1exp(Z/a)+b1exp(−Z/a), ρ2(Z) = b+a2exp(Z)+b2exp(−Z), (12)

and we define . Clearly, symmetry considerations invite us to define as the middle plane of the layer of -phase material, so and , from which follow and . Instead of determining the remaining two free parameters by the criteria pursued in Long (), we add a crossing criterion at , with ,

 ρ1(L/2) = ρ1,×, ρ2(L/2) = qb, (13)

assuming that the width of the layer, , is sufficiently large for the second equation to possess a solution (i.e. ). Note that and are, for the time being, free parameters. This crossing criterion is quite different from the trajectory-intersection strategy proposed in Long (). In particular, the trajectory intersection in the model in Long () always occurs near the -phase point, while the crossing occurs, as we shall see, roughly half a wetting layer thickness “away” from that point. Further, the crossing criterion relates the remaining free parameters and to the parameters and , for an arbitrarily chosen value of , through the simple relations

 a1 = ρ1,×2sinh(L/2a), a2 = −b4cosh(L/2). (14)

The constrained order parameter solutions then take the simple analytic forms:

 ρ1(Z) = ρ1,×sinh(Z/a)sinh(L/2a), ρ2(Z) = b−(1−q)bcosh(Z)cosh(L/2). (15)

In the spirit of a “double parabola approximation” we now proceed to apply the harmonic approximation also about the -phase point in the -plane. This leads to

 F(ρ1,ρ2;a,b)≈F(2)(ρ1,ρ2;a,b)≡4(1a2+b2)[(ρ1−1)2+ρ22]. (16)

The Euler-Lagrange equations are solved by, with , where ,

 ρ1(Z′) = 1−c1exp(−Z′), ρ2(Z′) = c2exp(−Z′), (17)

where we already implemented the boundary conditions appropriate to the phase point, which must be reached for . The two approximate pairs of solutions, one valid near and the other valid near , can be matched at in the solutions (III), which corresponds to in the solutions (III). This implies

 c1 = 1−ρ1,×, c2 = qb. (18)

In this way, a requirement of continuity of the order parameters at the crossing point fixes the parameters and in terms of two remaining freedoms. While the order parameters are continuous, their derivatives need not be, at the matching point in the -plane.

Within this double-parabola approximation we now define, as usual, the interface potential as the constrained surface free energy of a layer of of finite thickness adsorbed at the interface, minus the surface free energy of an infinitely thick layer (the wet profile). The first part of the surface free energy cost is obtained by evaluating the functional (1) between the limits and in the solutions (III) and the second part is obtained by evaluating the functional (1) between the limits and in the solutions (III). Note that the second part is independent of . We write, suggestively,

 V(L)/2≡σ(2)β,[0,L/2]+σ(2)γ,[0,∞]−(σ(2)β,[0,∞]+σ(2)γ,[0,∞]), (19)

where denotes the functional evaluated in the optimal profiles. Calculation entails

 V(L)=√A(−(1−q)2b2e−L1+e−L+ρ21,×ae−L/a1−e−L/a), (20)

which has a hard-wall divergence for . On the other hand, the surface free energy of the wet profile corresponds to (twice) the interfacial tension of the interface. In this harmonic approximation this reads

 σ(2)βγ≡σ(2)β,[0,∞]+σ(2)γ,[0,∞], (21)

and we obtain

 σ(2)βγ=√A2((1−q)2b2+ρ21,×a)+√C2(q2b2+(1−ρ1,×)2). (22)

It is interesting to examine the precise form of the wet trajectory in the -plane. This trajectory consists of two parts. The first part, from the phase point till the crossing point, is most easily obtained by recasting the solutions (III) in a form which is suitable for starting from the phase point for and reaching the crossing point at ,

 ρ1(Z′) = ρ1,×exp(Z′/a), ρ2(Z′) = b(1−(1−q)exp(Z′)), (23)

which implies

 ρ2=b−(1−q)b(ρ1ρ1,×)a,forρ1≤ρ1,×. (24)

The second part of the wet trajectory follows directly from (III) and takes the form

 ρ2=qb1−ρ11−ρ1,×,forρ1≥ρ1,×, (25)

which is a straight line. It is interesting to note that the slope at jumps from to . Now requiring continuity of the slope of the wet trajectory at , allows us to eliminate one freedom. We thus obtain

 q=a(1−ρ1,×)ρ1,×+a(1−ρ1,×). (26)

Note that the non-wet trajectory will, in general, display a discontinuity in slope at , whose magnitude depends also on . A next opportunity to eliminate a free parameter is provided by asking that the interfacial tension be minimal with respect to , after substitution of (26). One readily checks that the minimum is reached for

 ρ1,×=(1+1+b22√1+a2b2)−1, (27)

which takes the value 2/3 at (for any ) and 0.623 at an endpoint of the line of infinite order transitions, and . In conclusion, adopting this criterion for fixing would lead to a value that is rather insensitive to the parameters and in the region of our interest. We shall see further that the physical results we will derive are largely independent of the precise value of . Figure 1 illustrates our harmonic approximations supplemented with the crossing criterion and provides a comparison with the numerically exact trajectories, for both non-wet and wet states.

From (20) we now derive the asymptotic form, for large , of the interface potential , in the domain appropriate to critical wetting,

 V(L)/√A=−(1−q)2b2e−L+ρ21,×ae−L/a+(1−q)2b2e−2L+O(e−2L/a,e−3L), (28)

with the (scaled) thickness of the wetting layer of phase intruding between phases and . We recall that is measured in units of the scaled distance and the factor of 2 absorbed in our definition of allows for the fact that the actual thickness of the wetting layer of corresponds, by symmetry, to about twice the “wetting layer thickness” defined in the calculations of Long (); note (). Note that this interface potential differs qualitatively from the one engineered in Long (). Our present derivation is simpler and more transparent. While the difference is irrelevant at the level of the mean-field results due to the precise dependence on of the coefficient of the third term in (i.e., the mean-field critical exponents are not modified), the difference can become important at the level of the RG calculations.

Let us briefly derive the mean-field critical exponents from (28). Clearly, for we retrieve consistency with a first-order wetting transition, because the leading term then becomes positive. For critical wetting is possible. For a universal second-order wetting transition is retrieved from the interface potential

 V(L)√A=(−(1−q)2b2+ρ21,×)e−L+((1−q)2b2+ρ21,×)e−2L+O(e−3L), (29)

which predicts that, in this harmonic approximation scheme, complete wetting takes places for less than the threshold

 b(2)w(1)=ρ1,×/(1−q)=1, (30)

using (26) at , while the exact value is KogaWidom2008 (). Here we see explicitly that the value of the free parameter in the crossing criterion affects the theory at the level of non-universal properties such as the wetting transition phase boundary. For example, if (23) is adopted for , then solves (29).

For , (28) predicts that a critical wetting transition is possible when the leading coefficient, , which denotes the “distance” to the critical wetting phase boundary, tends to zero, so . The structure of the full interface potential (20) implies that critical wetting does indeed take place within the double-parabola approximation to the model. The equilibrium wetting layer thickness diverges logarithmically in the manner

 ^L∼2a1−a[ln1b+ln1a+const.], (31)

for . The spreading coefficient is predicted to vary as

 −S∝b2/(1−a), (32)

implying

 2−αs=11−a, (33)

in full agreement with what was already analytically conjectured and numerically verified in Long (). Interestingly, as was also anticipated in Long (), there is no cross-over to a universal second-order wetting transition at . Indeed, this is conspicuous and emerges naturally in our new derivation, because the coefficient of varies as , so that the third term of near wetting behaves as , which is always sub-dominant with respect to the second term, no matter how small is taken. Recall that the second as well as the first term near wetting behave as . This persistent non-universality is confirmed by considering the full given in (20) and is in excellent agreement with the prediction of high-precision computations in Long () that critical wetting remains non-universal from down to at least. Intriguingly, one of the implications of this theory is that critical wetting is of order less than 2 for and asymptotically becomes first order in the limit , an exotic scenario already sketched in Long ().

For the infinite-order wetting transition, emerging in the limit , we retrieve precisely the predictions of Long (),

 ^L∝11−a, (34)

implying , and

 −S∝⎡⎣bb(2)w(1)⎤⎦2a/(1−a)∝e−C/(1−a),withC>0, (35)

confirming the infinite-order character of the transition. All these results are essentially independent of the numerical value given to , corroborating the robustness of the crossing criterion we adopted to derive an interface potential.

## Iv Thermal fluctuation effects on infinite-order wetting: linear functional RG approach

Within local Hamiltonian theory the RG approach starts by considering an effective interface Hamiltonian of the form

 H[ℓ]kT=∫dx∫dy{Ke2(∇ℓ(x,y))2+V(ℓ(x,y))}, (36)

where, since the wetting layer is bounded by two similar interfaces whose thermal fluctuations are independent and additive, the (dimensionless) effective surface tension satisfies the following combining rule KardarInd (); CM ()

 1Ke=kTξ2(1σαβ+1σβγ)≡kTξ22σ, (37)

with the bulk correlation length in the phase, and (owing to the symmetry of the model). Note that our definitions of and entail the following definition for the wetting parameter FH ()

 ω=14πKe=kT2πσξ2, (38)

which is twice the value (10) for a single interface unbinding from a flat wall. Therefore, we are dealing with significantly enhanced thermal fluctuations as compared to the usual SRCW problem in three dimensions!

We proceed in two stages. In the first stage, we keep only the leading terms in the large- expansion of , with , and discuss the resulting singular behaviour at critical wetting. In the second stage, we include - as is physically required FH () - a term which mimics a soft repulsion penalizing the two interfaces when attempting to cross each other (penalizing negative values of ). It would be reasonable physically to consider a hard repulsion, but the linear RG approach cannot properly handle an infinite potential. In the linear functional RG approach the renormalized interface potential is obtained by integrating out capillary wave fluctuations. This amounts to a convolution of the bare potential with a Gaussian of width , where is the roughness of the fluctuating interface BHL (); FH (). The width , often referred to as the perpendicular correlation length of the interface, , can be calculated using capillary wave theory CW1 (); CW2 (). It is related to the parallel correlation length of the interface through

 ξ⊥=√2ωlnξ∥, (39)

where all lengths are scaled with the bulk correlation length . After these general considerations we now focus first on the regime of infinite-order transitions and therefore assume and .

Stage 1. To alleviate the notation we model the bare potential as

 V(L)={−De−L+Be−L/a,forL>00,forL<0, (40)

with and positive constants near infinite-order wetting, which satisfy .

At this stage no care is taken to exclude those (rare) capillary wave fluctuations that would lead to crossings of the two wandering interfaces. Since the calculations closely follow those outlined in FH () we do not repeat them here, but limit ourselves to reporting the results and mentioning only those calculational details that merit special attention. The renormalized potential reads

 VR(L)=−Dξω∥e−L+Bξa−2ω∥e−L/a,for largeL, (41)

which has the same form as but with multiplicatively renormalized coefficients. We recall that all lengths are implicitly scaled with the bulk correlation length . This form of renormalized potential is valid under the following conditions: for the first term and for the second term. Since the latter condition implies the former, but note that for the conditions become coincident. This condition expresses that the two interfaces fluctuate far enough from each other to avoid mutual collisions and defines the so-called weak fluctuation regime. We will see shortly that this corresponds to a definite range of .

Minimization of leads to the renormalized equilibrium wetting layer thickness, which we denote by , and the second derivative of evaluated in provides the parallel correlation length through , as outlined in FH (). Combining the resulting relations and eliminating the dependence on the ratio , which is merely a constant upon approach of the infinite-order wetting transition, leads to the following relation between and ,

 ^LR=ln(1a−1)+(2+ω)lnξ∥+const., (42)

where we show only the terms that matter in the limits and . A similar calculation leads to the following relation between and , which we present in the form, valid for ,

 e−^LR≈(B/D)−(2+ω)/[(2−ω)(1a−1)][const.(1a−1)]2ω/(2−ω), (43)

which implies, asymptotically for ,

 ^LR∼2+ω(2−ω)(1a−1)lnBD−2ω2−ωln(1a−1). (44)

Note that the first term determines the leading algebraic divergence of , with a critical exponent that is unchanged with respect to the mean-field result. Also note that the amplitudes of both leading and subleading terms diverge for (strong thermal fluctuations).

We also derive the following relation between and , likewise valid for ,

 ξ∥≈(BD)1/[(2−ω)(1a−1)][const.(1a−1)]−1/(2−ω),withB/D>1. (45)

Note that the first factor captures the leading exponential divergence, for , and the second factor embodies an algebraic divergence of the amplitude of this singularity, for fixed .

The relation between , defined through (39), and merits our special attention. Indeed, at the infinite-order wetting transition displays a universal algebraic divergence of the form

 ξ⊥∝√2ω2−ω(1−a)−1/2, (46)

implying the following result for the critical exponent of the thermally fluctuating interface width

 ν⊥=1/2. (47)

This is interesting because, to our knowledge, for other SRCW transitions in , we invariably have (log). Note, once again, that the amplitude diverges for sufficiently strong fluctuations, i.e., for .

Note that displays an exponential singularity, while and show an algebraic divergence, in the limit . Therefore, taking twice the logarithm of (45) we can rewrite (42) in the more systematic form of an expansion in large ,

 ^LR=(2+ω)lnξ∥−lnlnξ∥+... (48)

This allows us to check the self-consistency requirement, , for ,

 ^LR/lnξ∥≈2+ω>(2/a)ω, (49)

which leads to the following condition on ,

 ω<2(2/a)−1≈2. (50)

We conclude that there is only one “weak” (and no “intermediate”) fluctuation regime for infinite-order wetting, defined by . Note that for an additional exponential singularity develops in , implying a wetting transition of doubly-infinite order. The “strong” fluctuation regime, , that lies beyond this threshold appears interesting, too, but falls outside the scope of our paper.

We proceed to examine the leading singularity of the spreading coefficient at infinite-order wetting in the presence of thermal fluctuations and to check hyperscaling. Evaluating the interface potential at the equilibrium wetting layer thickness leads to

 −S≈(BD)−2/[(2−ω)(1a−1)][const.(1a−1)]2/(2−ω),withB/D≈[b(2)w(1)/b]2>1. (51)

We observe that , which implies that hyperscaling holds. (We recall that hyperscaling amounts to the exponent equality .) It can readily be seen that hyperscaling holds quite generally for the interface potentials of the type that we study, since the second derivative of is proportional to itself, when both are evaluated at .

We conclude that the main characteristics (the critical exponents) of the mean-field infinite-order wetting transition ( and ) are robust to thermal fluctuations, i.e., independent of the value of , provided . In addition, we remark that the power of in the argument of the exponential singularity (i.e., in the exponent of in (51)) is also robust to thermal fluctuations and preserves its mean-field value , as in (35), whereas the amplitude of in this argument does depend on .

Stage 2. We now progress towards a physically better founded model and add a non-zero soft repulsion between the two fluctuating interfaces by augmenting the bare interface potential with a positive constant at ;

 V(L)={−De−L+Be−L/a,forL>0E,forL<0, (52)

Since the constant renormalizes (approximately) to a Gaussian, the renormalized potential now reads, for ,

 VR(L)=−Dξω∥e−L+Bξa−2ω∥e−L/a+EL√ωlnξ∥πe−L2/(4ωlnξ∥),for largeL (53)

We must now check whether the solution for obtained at stage 1 is still valid. This is the case provided the Gaussian remains small compared to the other two terms, when the solution found at stage 1, being (48), is inserted in . One verifies that the first term scales as , the second as , with close to the wetting transition, and the third term scales as . Since for all (equality for ), the third term is negligible compared to the other two for . In conclusion, the properties of the (renormalized) infinite-order wetting transition are not sensitive to whether or not a soft repulsion is added to the interface potential. Calculational stages 1 and 2 are equivalent for this particular wetting transition.

## V Thermal fluctuation effects on non-universal wetting (a<1): linear functional RG approach

In this Section we renormalize the wetting transition of continuously varying finite order. We thus concentrate on the parameter ranges and . Recall that now . In doing so we will reproduce the results of Hauge and Olaussen (HO), who pioneered RG corrections to a wetting transition that has non-universal character at mean-field level HO (). They studied the weak fluctuation regime. Parry et al. bern () also examined the intermediate fluctuation regime for that transition and found that the universality with respect to asymmetry is restored, while the non-universality with respect to the wetting parameter persists. Why do we revisit these non-universalities, given that they have been explored before? The interface potential we encounter in our theory is qualitatively different from those of previous works, in that the coefficients of powers of all vanish at wetting, while the coefficients of powers of do not. This has important consequences for the size of the non-universal regime in the wetting phase diagram. Further, we pay special attention to the fact that in our case of two fluctuating interfaces thermal fluctuation effects are significantly enhanced, since - as we outlined in the previous section - the value of is roughly doubled relative to that for wetting at a planar undeformable wall. As in the previous section, we proceed in two calculational stages or “models”. At stage 1 we consider only the leading terms, for large , in the interface potential, and at stage 2 we add a soft repulsion to discourage interface wandering from visiting the unphysical domain .

Stage 1. We start from the bare interface potential based on (28)

 V(L)={−De−L+Be−L/a+De−2L,forL>00,forL<0, (54)

with the “distance” to the critical wetting transition at , and a positive constant.

Its renormalized counterpart is given by

 VR(L)=−Dξω∥e−L+Bξa−2ω∥e−L/a+Dξ4ω∥e−2L,for largeL, (55)

and is valid in the weak fluctuation regime , where is to be calculated. This form of renormalized potential is valid under the following conditions: for the first term, for the second term and for the third term. Note that, for below some value, the next-to-leading term in may “cross” some higher-order term and the transition may lock in to one of the universal critical wetting kind. At mean-field level, however, this does not happen for our model (see Section III). In order to exclude any possible complications of this sort, we limit ourselves for the time being to the range , where is to be calculated. It then suffices to work with the first two terms in (55) to derive the singularities at wetting.

Minimization of the first two terms of leads to the following relation between the (diverging) equilibrium wetting layer thickness and the (diverging) quantities and ,

and, combined with the evaluation of the second derivative, or “curvature”, in the minimum of , this provides an independent relation between the parallel correlation length and, essentially, the “field” by which wetting can be induced in the limit ,

These two auxiliary equations, (56) and (57), now provide the asymptotic relation between and close to wetting,

 ^LRlnξ∥≈2a+ω/a, (58)

The sufficient condition for the validity of the first two terms in (55), being , requires this ratio (58) to exceed , and signifies that the interfaces fluctuate sufficiently far from one another to avoid collisions. It defines the weak fluctuation regime,

 ω<2a2≡ω1 (59)

As expected, we retrieve the same as in the model studied by HO, which further reduces, for , to the famous value 1/2 found in FH ().

We proceed to study the critical exponent of the correlation length parallel to the unbinding interfaces. Defining, as is standardly done,

 ξ∥∝D−ν∥, (60)

and using (57) we find

 ν∥=12(1−a)11−ω/(2a), (61)

as in the model of HO HO (), and consequently we obtain log), in view of (39). The first factor in the r.h.s. of (61) defines the mean-field contribution (the limit ) and last factor contains the thermal fluctuation correction. Note that the order of the transition increases from its mean-field value when is increased, i.e., when thermal fluctuations gain importance. One easily verifies, along the lines of the derivation given in FH (), that hyperscaling holds so that .

We now show that in the regime of validity of the first three terms in (58), defined by the sufficient condition and , i.e., and , the third term evaluated in is negligible compared to the first two, not only for but for all , so . Indeed, using (58) and (60) to examine how the various terms in scale near wetting, we obtain that the first two terms scale as while the third term scales as , which is subdominant provided . This is indeed fulfilled, since the former condition implies this one for all .

The question now remains whether also holds for but , so that the third term in (55) must be replaced by a suitably renormalized version of the bare term . This question is obviously only meaningful for , since for the validity of the second term implies that of the third and we are back to the previous conclusion reached in case all three terms in (55) are valid. The suitably renormalized interface potential now reads, the third term being a Gaussian (with a positive amplitude) FH (),

 VR(L)=−Dξω∥e−L+Bξa−2ω∥e−L/a+D√4πωlnξ∥12−L/(2ωlnξ∥)e−L2/(4ωlnξ∥)+...,