Interfaces and wetting transition on the half plane. Exact results from field theory
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r SISSA 07/2013/FISI
Interfaces and wetting transition on the half plane.Exact results from field theory
Gesualdo Delfino and Alessio Squarcini
SISSA – Via Bonomea 265, 34136 Trieste, ItalyINFN sezione di Trieste
Abstract
We consider the scaling limit of a generic ferromagnetic system with a continuous phasetransition, on the half plane with boundary conditions leading to the equilibrium of twodifferent phases below criticality. We use general properties of low energy two-dimensionalfield theory to determine exact asymptotics of the magnetization profile perperdicularly tothe boundary, to show the presence of an interface with endpoints pinned to the boundary,and to determine its passage probability. The midpoint average distance of the interfacefrom the boundary grows as the square root of the distance between the endpoints, unlessthe reflection amplitude of the bulk excitations on the boundary possesses a stable boundstate pole. The contact angle of the phenomenological wetting theory is exactly related tothe location of this pole. Results available from the lattice solution of the Ising model arerecovered as a particular case.
Introduction
Interfacial phenomena at boundaries are a subject of relevant interest for both theory andapplications. On the theoretical side, the one this paper is concerned with, the effects of theboundary on an interface separating different phases of a statistical system have been extensivelystudied using phenomenological, mean field, renormalization group and other approximationmethods ([1-8] is a certainly incomplete list of review articles). The only exact result that hasbeen available concerns the Ising model on the half plane [9, 10], a circumstance that, whileconfirming a specificity of the two-dimensional case, raises the question about the role of Isingsolvability in these exact findings.We show in this paper that exact results including those of [10] as a particular case areobtained quite generally for any two-dimensional model exhibiting a continuous phase transition.This is done extending to the half plane the non-perturbative field theoretical approach recentlyused in [11] to study phase separation on the whole plane. As in that case, general exact resultsemerge because, when its end-to-end distance R is much larger than the correlation length, theinterface is described by a single particle (domain wall) state, in a low energy limit leading to ageneral solution. In this way, the fluctuations of the interface turn out to be ruled by the lowenergy singularity of the matrix element of the order parameter field (as for the whole plane),with the fields pinning the interface endpoints to the boundary producing boundary reflectionand an average midpoint distance from the boundary of order √ R .The result changes qualitatively if boundary and domain wall excitation admit a stable boundstate, which becomes dominant in the spectral sum at low energies and bounds the interface tothe boundary. The contact angle and the spreading coefficient of the phenomenological theoryof wetting then emerge in a completely natural way within the field theoretical formalism.The paper is organized as follows. In the next section we illustrate the field theoreticalsetting and derive the results for the unbound interface. Section 3 is then devoted to the effectsproduced by the bound state and to the characterization of the wetting transition, while section 4contains some final remarks. Consider a ferromagnetic spin model of two-dimensional classical statistical mechanics in whichspins take discrete values labelled by an index a = 1 , , . . . , n . The energy of the system isinvariant under global transformations of the spins according to a symmetry whose spontaneousbreaking below a critical temperature T c is responsible for the presence on the infinite plane of n translation invariant pure phases; we denote h· · · i a statistical averages in the phase a .Assuming a continuous transition, we consider the scaling limit below T c , corresponding toa Euclidean field theory defined on the plane with coordinates ( x, y ), which can be seen as theanalytic continuation to imaginary time of a (1+1)-dimensional relativistic field theory withspace coordinate x and time coordinate t = iy . If H and P are the Hamiltonian and momentum1 ba (a) (b) Figure 1: Elastic scattering (reflection) of a kink off the boundary (a), and interface pinned atthe boundary (b).operators and Φ a field of the theory, translation invariance on the plane yields the relationΦ( x, y ) = e ixP + yH Φ(0 , e − ixP − yH . (1)The (1+1)-dimensional theory possesses degenerate vacua | i a associated to the pure phases ofthe system. The elementary excitations correspond to stable kink states | K ab ( θ ) i interpolatingbetween different vacua | i a and | i b . We introduced the rapidity variable θ which convenientlyparameterizes the energy and momentum of the kinks as ( E, p ) = ( m cosh θ, m sinh θ ), m beingthe kink mass or inverse correlation length. The trajectory of the kink on the Euclidean planecorresponds to a domain wall between the phases a and b . Multi-kink excitations take the form | K aa ( θ ) K a a ( θ ) . . . K a n − b ( θ n ) i . Within the scattering framework [12] we consider, these areasymptotic states, incoming if considered long before the collisions among the kinks, outgoingif considered long after, and their energy is simply P ni =1 m cosh θ i .Consider now the system on the half-plane x ≥
0. We denote by B a a boundary conditionat x = 0 which is y -independent and breaks the symmetry of the bulk in the direction a in orderparameter space; this can be realized applying a constant boundary magnetic field pointing inthe direction a . We denote h· · · i B a statistical averages in presence of the boundary condition B a . Preservation of translation invariance in the y direction yields energy conservation in the(1 + 1)-dimensional picture. The bulk excitations are still the kink states described for the fullplane case, but now they are restricted to x >
0; we indicate this restriction by a subscript B a . Hence | i B a denotes the vacuum (no excitations in the bulk) on the half-plane with theboundary condition B a . If σ is the spin field, the magnetization h σ ( x, y ) i B a = B a h | σ ( x, y ) | i B a points in the direction a and depends only on the distance x from the boundary; in particularlim x →∞ h σ ( x, y ) i B a = h σ i a , (2)where h σ i a is the constant magnetization in phase a on the full plane. The state | i B a iseigenstate of the Hamiltonian H B a of the system on the half line. We consider the case in which2oundary conditions B a and B b are related by the symmetry, so that | i B a and | i B b have thesame energy E B .The asymptotic scattering state | K ba ( θ ) i B a corresponds to an incoming kink (travelling to-wards the boundary) if its momentum is negative, i.e. if θ <
0. If its energy is lower than theenergy 2 m needed to produce two kinks upon interaction with the boundary, it will simply bereflected into an outgoing kink with rapidity − θ (Fig. 1a). The state | K ba ( θ ) i B a is eigenstateof H B a with eigenvalue E B + m cosh θ .We are now ready to set up the configuration we want to study, namely a boundary conditionwhich is of type B a if | y | > R/ B b if | y | < R/
2. The interest of such a boundarycondition, that we denote B aba , is easily understood observing that the limit for x → ∞ of themagnetization profile h σ ( x, i B aba has to tend to h σ i a if R is finite, and to h σ i b if R is infinite.The natural way to account for this situation is to expect the formation of an interface pinnedat R/ − R/ b from an outer phase a (Fig. 1b),and whose average distance from the boundary at y = 0 diverges with R . The remainder of thissection is devoted to see how such a picture indeed emerges within our general field theoreticalframework.Technically the change from the boundary condition B a to B b at a point y is realized startingwith B a and inserting on the boundary a field µ ab (0 , y ) which acting on the vacuum | i B a createskink states interpolating between phase a and phase b . Hence the simplest non-vanishing matrixelement of the boundary field µ ab is B a h | µ ab (0 , y ) | K ba ( θ ) i B a = e − ym cosh θB a h | µ ab (0 , | K ba ( θ ) i B a ≡ e − ym cosh θ F µ ( θ ) . (3)The partition function of the system with boundary condition B aba reads Z = B a h | µ ab (0 , R/ µ ba (0 , − R/ | i B a = Z ∞ dθ π |F µ ( θ ) | e − mR cosh θ + O ( e − mR ) , (4)where the last expression is obtained expanding over an intermediate set of outgoing kink statesand retaining only the lightest (single kink) contribution which is leading in the large mR limitwe will consider from now on. Since the above integral is dominated by small rapidities and F µ is expected to behave as F µ ( θ ) = a θ + O ( θ ) , (5)the partition function becomes Z ∼ | a | Z ∞ dθ π θ e − mR (1+ θ / = | a | e − mR √ π ( mR ) / . (6) As emphasized in [13], the analogies between bulk and boundary scattering become evident thinking of theboundary as the propagation of an infinitely heavy particle sitting at x = 0. Linear behavior of matrix elements at small rapidities in two-dimensional theories is well known. Within theframework of integrable boundary field theory [13] exact examples can be found in [14]. More generally, see [15]about matrix elements in integrable theories. x axis is given by h σ ( x, i B aba = 1 Z B a h | µ ab (0 , R/ σ ( x, µ ba (0 , − R/ | i B a (7) ∼ Z Z + ∞−∞ dθ π dθ π F µ ( θ ) h K ab ( θ ) | σ (0 , | K ba ( θ ) iF ∗ µ ( θ ) e m [ i (sinh θ − sinh θ ) x − (cosh θ +cosh θ ) R ] , where in the last line we have taken mR ≫ mx ≫ σ ( x,
0) as a bulk field which satisfies (1) and is evaluated on bulkkink states (whose rapidities take both positive and negative values). In other words, for mx large the only effect of the boundary on the magnetization comes from the boundary changingfields at (0 , ± R/ h σ i a . The bulkmatrix element of the spin field between one-kink states is related by the crossing relation h K ab ( θ ) | σ (0 , | K ba ( θ ) i = F σ ( θ + iπ − θ ) + 2 πδ ( θ − θ ) h σ i a , (8)to the form factor F σ ( θ − θ ) ≡ a h | σ (0 , | K ab ( θ ) K ba ( θ ) i . (9)As already observed in [11] for the case of phase separation on the whole plane, it is crucial thatquite generally, due to non-locality of the kinks with respect to the spin field, F σ ( θ ) possessesan annihilation pole at θ = iπ with residue [16] − i Res θ = iπ F σ ( θ ) = h σ i a − h σ i b ≡ ∆ h σ i . (10)Since mR is large (7) is dominated by small rapidities and (5), (8) and (10) lead to h σ ( x, i B aba ∼ h σ i a + i ∆ h σ i | a | Z e − mR Z + ∞−∞ dθ π dθ π θ θ θ − θ e m [ i ( θ − θ ) x − ( θ + θ ) R ] . (11)Differentiation removes the singularity of the integrand and gives ∂ mx h σ ( x, i B aba ∼ − ∆ h σ i | a | e − mR (2 π ) Z g ( x ) g ( − x )= ∆ h σ i √ √ π mR z e − z , z ≡ r mR x (12)where we used (6) and g ( x ) = Z + ∞−∞ dθ θ e − mRθ / imxθ = 2 i √ πmR z e − z / . (13)Integrating (12) with the asymptotic condition h σ ( ∞ , i B aba = h σ i a gives h σ ( x, i B aba ∼ h σ i b − √ π ∆ h σ i (cid:18) z e − z − Z z du e − u (cid:19) , mx ≫ . (14) Crossing a particle from the initial to the final state (or vice versa) involves reversing the sign of its energyand momentum [12], namely an iπ rapidity shift. The delta function term in (8) is a disconnected part arisingfrom annihilation of the two kinks. R →∞ h σ ( αm ( mR ) δ , i B aba , obtaining h σ i b for 0 < δ < / h σ i a for δ > /
2, and the r.h.s. of (14) with z = α √ δ = 1 /
2. For h σ i a = −h σ i b = h σ i + these are precisely the limits obtained from the lattice in [17, 9] for the Ising model on the halfplane with boundary spins fixed to be positive for | y | > R/ | y | < R/ z = 1, confirming thepresence of an interface whose average distance from the boundary increases as p R/m . It is alsoeasy to see that the result for the magnetization profile is consistent with a simple probabilisticinterpretation. Since we are computing the magnetization on a scale R much larger than thecorrelation length and far away from the boundary, we can think of the interface as a sharpseparation between pure phases , and write h σ ( x, i B aba ∼ h σ i a Z x du p ( u ) + h σ i b Z ∞ x du p ( u ) , mx ≫ , (15)where p ( u ) du is the probability that the interface intersects the x -axis in the interval ( u, u + du ), so that the two integrals are the left and right passage probabilities with respect to x .Differentiating and comparing with (12) gives the passage probability density p ( x ) = 4 r mπR z e − z , (16)which correctly satisfies R ∞ dx p ( x ) = 1. The results of the previous section are modified if the kink-boundary system associated to theasymptotic state | K ab ( θ ) i B b admits a stable bound state | i B ′ a , corresponding to the bindingof the kink K ab on the boundary B b . As usual for stable bound states [12], such a bindingwill correspond to a “virtual” value θ of the kink rapidity leading to a bound state energy E B + m cosh θ real and smaller than the unbinding energy E B + m . This amounts to taking θ = iu with 0 < u < π , so that E B ′ = E B + m cos u . (17)The existence of the bound state manifests in particular through a simple pole in the elasticscattering amplitude of the kink off the boundary, which reads R ( θ ) ∼ ig / ( θ − iu ) for θ → iu ,with g a kink-boundary coupling constant (Fig. 2a). This pole is inherited by the matrix element(3), for which we have (Fig. 2b) F µ ( θ ) = B a h | µ ab (0 , | K ba ( θ ) i B a ∼ igθ − iu B a h | µ ab (0 , | i B ′ a , θ → iu . (18) It has been shown in [11] how the internal structure of the interface arises from subleading terms in the large mR expansion. Exact solutions exhibiting boundary bound states poles can be found in [13] for scattering amplitudes and in[14] for matrix elements. ab b a µ ab gg abb ~ (a) (b) g b a Figure 2: The boundary bound state (double line) originating in kink-boundary scattering (a),and a pictorial representation of equation (18) (b).The boundary bound state affects the results of the previous section for the boundary condi-tion B aba because the leading low-energy contribution in the expansion over intermediate statesnow comes from | i B ′ a rather than from | K ba ( θ ) i B a . So the partition function becomes Z = B a h | µ ab (0 , R/ µ ba (0 , − R/ | i B a = (cid:12)(cid:12) B a h | µ ab (0 , | i B ′ a (cid:12)(cid:12) e − mR cos u + O ( e − mR ) , (19)and the magnetization profile h σ ( x, i B aba ∼ Z B a h | µ ab (0 , R/ | i B ′ a B ′ a h | σ ( x, | i B ′ a B ′ a h | µ ba (0 , − R/ | i B a = h σ ( x, i B ′ a . (20)We see then that, as a consequence of (2), the magnetization profile now tends to h σ i a at large mx , in contrast to what obtained in the previous section, where it tended to h σ i b for R largeenough. This corresponds to the fact that now the asymptotic behavior is determined by thestate in which the interface, and then the phase b , are bound to the boundary, while before thedominant state was that in which phase b extended to an average midpoint distance of order √ R from the boundary.Consistency of the asymptotic expansion requires that the corrections to (20) vanish as R → ∞ . For mx large, the first of these corrections is that due to the | K ba ( θ ) i intermediatestates given in (7). The Z in the denominator, however, is now (19) rather than (6), so thatthe correction behaves as e mR (cos u − at large R . Hence, if u approaches 0, i.e. if the interfaceapproaches the unbinding point, consistency requires that R diverges faster than 1 /u . If weadopt a vocabulary within which b is a liquid phase and a a vapor phase, we can say that as u → b θ e Figure 3: Splitting and recombination of the boundary bound state B ′ a corresponds to “partialwetting”, in which a drop of phase b makes an equilibrium contact angle θ e with the boundary.Equation (17) with u = θ e gives the surface tension balance condition at the contact points.the boundary (see e.g. [8]). In our formalism this amounts to splitting and recombination of theboundary bound state B ′ a (Fig. 3). Considering that the kink mass m is the surface tension ofthe interface [11], that E B is the surface tension between the boundary and the drop, and that E B ′ is the surface tension between the boundary away from the drop and phase a , we recognizein (17) the Young equilibrium condition at contact points (see e.g. [1] and references therein),with u playing the role of the equilibrium contact angle θ e (Fig. 3). In addition, the combination m (cos u −
1) encountered a moment ago is recognized as the so called “equilibrium spreadingcoefficient” (see [8]). We also see that interface unbinding at u = 0 corresponds to vanishing ofthe contact angle, namely to the usual characterization of the wetting transition point (passagefrom partial to complete wetting).The boundary bound state is a property of the theory with translationally invariant boundarycondition B b . Parameters of this theory are the temperature, related to the kink mass as m ∝ ( T c − T ) ν , and a coupling λ entering the boundary term λ R dy φ (0 , y ) of the classicalreduced Hamiltonian. If X is the scaling dimension of the boundary field φ (0 , y ), u is functionof the dimensionless combination λ/m − X . If λ is kept fixed, the condition u = 0 determines awetting transition temperature T w ( λ ) < T c .The results (6), (19) and (20) account for those reported in [9, 10] for the particular case ofan Ising model with boundary condition B + − + and coupling between the boundary spins andtheir nearest neighbors different from the coupling within the rest of the lattice; this modifiedcoupling corresponds to the boundary parameter λ in this case. The generality of our resultsalso explains why approximated treatments of other models resulted in findings similar to theIsing ones (see [10] and references therein). The exponents ν and X are known exactly from bulk [18] and boundary [19] conformal field theory, respec-tively. Conclusion