week ending 8 AUGUST 2014
PHYSICAL REVIEW LETTERS
PRL 113, 066101 (2014)
Phase Separation in a Wedge: Exact Results Gesualdo Delfino* and Alessio Squarcini† SISSA—International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy and INFN—Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34127 Trieste, Italy (Received 14 March 2014; published 4 August 2014) The exact theory of phase separation in a two-dimensional wedge is derived from the properties of the order parameter and boundary condition changing operators in field theory. For a shallow wedge we determine the passage probability for an interface with endpoints on the boundary. For generic opening angles we exhibit the fundamental origin of the filling transition condition and of the property known as wedge covariance. DOI: 10.1103/PhysRevLett.113.066101
PACS numbers: 68.08.Bc, 11.10.Kk, 68.35.Rh
Interfacial phenomena at boundaries are a subject of both experimental and theoretical relevance which has received continuous and extensive interest in the last decades [1–4]. An aspect particularly important for applications is that the structure and geometry of the substrate can alter the adsorption characteristics of a fluid in an important way (see [5] for a review). Adsorption measurements can then be used, for example, to characterize fractally rough surfaces [6], or the connectivity of porous substrates [7]. The basic case of a wedge-shaped substrate [8] acquired special interest since phenomenological and thermodynamic arguments [9] indicated a specific relation with the adsorption properties of a completely flat surface: the wedge wetting (or filling) transition occurs at the temperature for which the contact angle of a fluid drop on a flat substrate equals the tilt angle α of the wedge, a circumstance that allows us to regulate the transition temperature adjusting α. The connections between adsorption characteristics for different opening angles are known as properties of wedge covariance [10–12] and are experimentally testable [13]. It is clearly important to pass from a macroscopic to a fundamental statistical mechanical description. In two dimensions the essential role of fluctuations was established by the exact lattice results for the Ising model on the half plane [14–19], which provided essential support for heuristic statistical descriptions of the wetting of a flat boundary [20]. For the wedge geometry the existence of the filling transition was established for the Ising model on a planar lattice forming a right-angle corner [21,22] (see also [23] for a 60° opening angle on the triangular lattice), but otherwise the theoretical investigations in two and three dimensions have been based on effective interfacial Hamiltonian models [10–12,24–26] or density functional methods [27]. In this Letter, we derive general exact results for phase separation in a two-dimensional wedge. This is achieved by exploiting low energy properties of bulk two-dimensional field theory [28,29] together with a characterization of the 0031-9007=14=113(6)=066101(5)
operators responsible for the departure of an interface from a point on a boundary. For a shallow wedge we determine the exact passage probability of an interface with endpoints on the boundary. The theory provides a fundamental meaning to the contact angle and, for generic α, yields the filling transition condition. More generally, wedge covariance is shown to originate from the properties of the boundary condition changing operators in momentum space. We begin the derivation considering a two-dimensional system at a first order transition point, close enough to a second order transition to allow a continuous description in terms of a Euclidean field theory on the plane with coordinates (x, y). We label by an index a ¼ 1; …; n the different coexisting phases, denote by σðx; yÞ the order parameter operator, and by hσia its expectation value in phase a. The Euclidean field theory corresponds to the continuation to imaginary time t ¼ iy of a relativistic quantum theory in one space dimension, for which phase coexistence amounts to the presence of degenerate vacua jΩa i associated to the different phases of the system. The elementary excitations are kink states jK ab ðθÞi interpolating between different vacua jΩa i and jΩb i; the rapidity θ parametrizes the energy and momentum of these particles as p0 ¼ mab cosh θ and p1 ¼ mab sinh θ, respectively, the kink mass mab being inversely proportional to the correlation length. The relativistic dispersion relation ðp0 Þ2 − ðp1 Þ2 ¼ m2ab is preserved by Lorentz boosts M Λ 11 with matrix elements [30] M00 Λ ¼ M Λ ¼ cosh Λ and 01 10 MΛ ¼ MΛ ¼ sinh Λ, which shift rapidities by Λ; as a consequence, relativistic invariant quantities depend on rapidity differences. If H and P are the Hamiltonian and momentum operators of the quantum system, a local operator Φ satisfies Φðx; yÞ ¼ eixPþyH Φð0; 0Þe−ixP−yH . We want to consider the system in the wedge geometry of Fig. 1, where the points (0, R=2) are boundary condition changing points. More precisely, a boundary field points in direction a (respectively, b) in order parameter space for jyj > R=2 (respectively, jyj < R=2)
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© 2014 American Physical Society
week ending 8 AUGUST 2014
PHYSICAL REVIEW LETTERS
PRL 113, 066101 (2014)
F μα ðθÞ ¼ F μ0 ðθ þ iαÞ:
y α
ð2Þ
The order parameter in the wedge for jyj < R=2 reads R/2
b 0
hσðx; yÞiW aba ¼ α
a
hΩa jμab ð0; R2 Þσðx; yÞμba ð0; − R2 ÞjΩa i−α ; Z W aba ð3Þ
x where jΩa iα ¼ M −iα jΩa i0 and
−R/2
Z W aba ¼ α hΩa jμab ð0; R=2Þμba ð0; −R=2ÞjΩa i−α :
α FIG. 1 (color online). Wedge geometry with boundary condition changing points at (0, R=2) and an interface running between them.
on the wedge, inducing a symmetry breaking in the bulk. If we denote by a subscript W aba the statistical averages in the wedge geometry with these boundary conditions, the limit for x → ∞ of the order parameter hσðx; yÞiW aba is expected to tend to hσia if R is finite, and to hσib if R is infinite. For mab R large this should correspond to an interface running between the points (0, R=2), separating an inner phase b from an outer phase a (Fig. 1), and whose average midpoint distance from the wedge diverges with R. To see how this emerges from the theory, consider first the case of a completely flat boundary (tilt angle α ¼ 0). With a uniform boundary condition of type a on the whole boundary we have a theory defined on the half plane x ≥ 0 with a vacuum state that we denote jΩa i0 . The change of boundary condition at the point (0, R=2) then corresponds to the insertion of an operator μab ð0; R=2Þ responsible for the emission from the boundary of kinks interpolating between phase a and phase b. The simplest matrix element of such a boundary operator is R jK ba ðθÞi0 ¼ e−mðR=2Þ cosh θ F μ0 ðθÞ; 0 hΩa jμab 0; 2
ð1Þ
We expand Eq. (4) by inserting a complete set of asymptotic states between the two boundary condition changing operators, take the limit of large mR in which the lightest (single kink) intermediate state dominates, and obtain Z ∞ dθ μ 2 Z W aba ∼ F α ðθÞF μ−α ðθÞe−mR½1þðθ =2Þ : ð5Þ 0 2π An emission amplitude F μ0 ð0Þ ≠ 0 would mean that the kink has a probability of remaining stuck on the boundary, a possibility that we are excluding for the time being. Hence, on general analyticity grounds [31], we have F μ0 ðθÞ ¼ cθ þ Oðθ2 Þ, so that Eq. (2) gives F μα ðθÞ ∼ cðθ þ iαÞ;
jθj; jαj ≪ 1;
ð6Þ
c2 e−mR ð1 þ mRα2 Þ: Z W aba ∼ pffiffiffiffiffiffi 2 2π ðmRÞ3=2
ð7Þ
and then
In a similar way, we evaluate Eq. (3) expanding over intermediate states and taking the limits mR ≫ 1, m½ðR=2Þ − jyj ≫ 1, which project onto the single kink states. We then obtain hσðx; yÞiW aba ×
where we set mab ≡ m; jK ba ðθÞi0 is an asymptotic kink state (outgoing if θ > 0) on the half plane and has energy E0 þ m cosh θ, E0 being the energy of the vacuum jΩa i0 . If we now move to a new frame through the Lorentz boost M Λ , the kink rapidity becomes θ þ Λ, and the boundary moves in time. For a purely imaginary Λ ¼ −iα the boost corresponds to a rotation in the Euclidean plane, and to a boundary forming an angle α with the y axis. By relativistic invariance the emission amplitude F μα ðθ − iαÞ of the kink with rapidity θ − iα from the moving boundary coincides with F μ0 ðθÞ, something that we can rewrite as
ð4Þ
e−mR ∼ Z W aba
Z
þ∞
−∞
dθ1 dθ2 μ F α ðθ1 ÞMσ ðθ1 jθ2 Þ ð2πÞ2
2 2 F μ−α ðθ2 Þe−ðm=2Þ½ððR=2Þ−yÞθ1 þððR=2ÞþyÞθ2 þimxðθ1 −θ2 Þ ;
Mσ ðθ1 jθ2 Þ ≡ hK ab ðθ1 Þjσð0; 0ÞjK ba ðθ2 Þi;
ð8Þ ð9Þ
evaluating the matrix element, Eq. (9), on bulk kink states we imply that the leading boundary effects for large mR are accounted for by the boundary condition changing operators inserted at the points (0, R=2). In the small rapidity limit which gives the leading contribution to the integral in Eq. (8) the matrix element of the order parameter operator takes the form [28,29]
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Mσ ðθ1 jθ2 Þ ∼ −
iΔhσi þ 2πδðθ1 − θ2 Þhσia ; θ1 − θ 2
ð10Þ
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PHYSICAL REVIEW LETTERS
where Δhσi ≡ hσia − hσib . The delta function term is a disconnected part corresponding to the kink trajectory passing aside the operator without interacting; the pole comes from the fact that the operator sees different phases if the kink passes to its right or to its left. We use Eq. (6) to evaluate Eq. (8) for small positive α, but also differentiate with respect to x in order to get rid of the pole in Eq. (10). The result is pffiffiffi m 3=2 ðx þ Rα ∂ x hσðx; yÞiW aba Þ2 − ðαyÞ2 −χ 2 pffiffiffi 3 2 e ; ∼8 2 R Δhσi π κ ð1 þ mRα2 Þ ð11Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where we defined κ ¼ 1 − 4y2 =R2 , χ ¼ 2m=Rðx=κÞ. Integrating back over x with the asymptotic condition hσðþ∞; yÞiW aba ¼ hσia gives pffiffiffiffiffiffiffiffiffiffi 2 χ þ 2mR ακ −χ 2 hσðx;yÞiW aba ∼ hσib þ Δhσi erfðχÞ − pffiffiffi : e π 1 þ mRα2
FIG. 2 (color online). Passage probability density Pðx; yÞ=m for the interface, with Pðx; yÞ given by Eq. (11), mR ¼ 25 and α ¼ 0.04. The leftmost contour line corresponds to Pðx; yÞ ¼ 0, and then to the wedge. The required divergence of the passage probability density as the pinning points (0, R=2) are approached is also clearly visible.
Z
ð12Þ For α ¼ y ¼ 0 and hσia ¼ −hσib , this result coincides with that originally obtained in [17] from the exact lattice solution of the Ising model on the half plane. It was shown in [28,29,32] that the leading term in the large mR (small rapidity) expansion corresponds to a sharp separation between pure phases, with subsequent terms accounting for the internal interface structure. In the present geometry the order parameter for sharp phase separation can be written as Z hσðx; yÞiW aba ∼ hσia
x~
x
Z dvPðv; yÞ þ hσib
∞
dvPðv; yÞ;
x
ð13Þ where Pðv; yÞ is the probability that the interface passes in the interval (v, v þ dv) on the line of constant ordinate y, which intersects the wedge at x ¼ x~ ðyÞ. It follows that Pðx; yÞ coincides with Eq. (11). R ∞It can be checked that the consistency requirement x~ dxPðx; yÞ ∼ 1 implies pffiffiffiffiffiffiffi mRα ≪ 1. The result, Eq. (11), shows, in particular, that Pðx; yÞ vanishes for jyj ¼ x=α þ R=2, which in the small α approximation of the present computation are the coordinates of the wedge (x ≥ −Rα=2). This shows that the properties, Eqs. (2) and (6), that we identified in momentum space for the matrix element of the boundary condition changing operator indeed lead to an impenetrable wedge in coordinate space. A plot of Pðx; yÞ is shown in Fig. 2. The average midpoint position of the interface reads
x¯ ¼
∞
−Rα=2
dx x Pðx; 0Þ
rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi 2R πmR 2 1þ α þ OðmRα Þ : ¼ πm 2
ð14Þ
The results, Eqs. (11) and (12), apply to values of temperature [i.e., of bulk correlation length m−1 ∝ ðT c − TÞ−ν ] for which the kink state is the lightest one entering the spectral decomposition of Eq. (3). To discuss the situation in which this is not the case we start again from α ¼ 0. For temperatures below a certain threshold T 0 < T c the kink K ba ðθÞ may form with the boundary a bound state jΩ0a i0 with energy E00, in which the phase b forms a thin layer adsorbed on the boundary. As usual for stable bound states, this corresponds to a purely imaginary rapidity θ ¼ iθ0 of the kink, leading to a binding energy E00 − E0 ¼ m cos θ0
ð15Þ
smaller than m. This boundary bound state is now the lightest state contributing to Eq. (3) and produces an order parameter equal to hσia for mx ≫ 1, no matter how large R is [32]. Since m is the interfacial tension between the phases a and b [28], and E0 (respectively, E00 ) corresponds to the tension between the boundary and phase b (respectively, a), Eq. (15) identifies θ0 as the contact angle of the phenomenological wetting theory. The usual relation θ0 ðT 0 Þ ¼ 0 characterizing the wetting transition temperature T 0 corresponds to the unbinding threshold for the kink.
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PRL 113, 066101 (2014)
a
µab
derivation makes transparent the relation of wedge covariance with the relativistic nature of the kink excitations, and explains why this property has been difficult to establish in general within nonrelativistically covariant effective interfacial models. The passage probability for an interface with endpoints on the wedge has also been determined for small tilt angles in the unbound regime.
µab a
b
g b
FIG. 3 (color online). Illustration of Eq. (16). The kink emission amplitude exhibits a bound state pole corresponding to adsorption of phase b on the boundary.
Bound states manifest in matrix elements as poles in the physical region of kinematical variables [31]. For the matrix element, Eq. (1), this physical region corresponds to the strip Im θ ∈ ð0; πÞ, and the pole induced by the boundary bound state takes the form (see Fig. 3) F μ0 ðθÞ ¼ 0 hΩa jμab ð0; 0ÞjK ba ðθÞi0 ig ∼ hΩ jμ ð0; 0ÞjΩ0a i0 ; θ − iθ0 0 a ab
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*
[1] [2]
[3]
[4] [5]
θ ∼ iθ0 ;
ð16Þ
[6] [7]
with g a coupling measuring the strength of the bound state. It then follows from Eq. (2) that the pole of F μα ðθÞ is located at θ ¼ iðθ0 − αÞ. Since the kink energy is always m cosh θ and the unbinding threshold remains at θ ¼ 0, the filling transition temperature T α is determined by the condition
[9] [10]
θ0 ðT α Þ ¼ α:
[11]
ð17Þ
For θ0 < α the pole is located at Im θ < 0, namely, outside the physical strip allowed for bound states; in such a case the kink is unbounded and phase b fills the wedge. The condition, Eq. (17), is that obtained in the macroscopic framework [9], and follows here from the exact statistical theory. Notice that while Eqs. (11) and (12) rely on Eq. (6), and then on small α, Eqs. (2) and (17) are general. Equation (2), in particular, encodes the essence of wedge covariance. For the scaling Ising model on the half plane with a boundary magnetic field h, the scattering amplitude off the boundary is known exactly [33], and exhibits a boundary bound state pole corresponding to 1 − sin θ0 ¼ h2 =ð2mÞ ¼ ðT c − T 0 ðhÞÞ=ðT c − TÞ; the last equality follows from νIsing ¼ 1 and holds in the scaling limit (see, e.g., [34] for boundary bound states in the lattice formalism). In summary, we constructed the exact theory of phase separation in a two-dimensional wedge and derived from it the filling transition condition and the origin of wedge covariance. This has been achieved directly in the continuum and for arbitrary opening angles, while previous exact results concerned only the lattice Ising model and were limited to a right-angle corner. In particular, our
[email protected] [email protected] P. G. de Gennes, Rev. Mod. Phys. 57, 827 (1985). S. Dietrich, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic Press, London, 1988), Vol. 12, p. 1. M. Schick, in Liquids at Interfaces, edited by J. Chavrolin, J.-F. Joanny, and J. Zinn-Justin (Elsevier, Amsterdam, 1990), p. 415. D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, Rev. Mod. Phys. 81, 739 (2009). S. Dietrich, M. N. Popescu, and M. Rauscher, J. Phys. Condens. Matter 17, S577 (2005). P. Pfeifer, Y. J. Wu, M. W. Cole, and J. Krim, Phys. Rev. Lett. 62, 1997 (1989). L. D. Gelb, K. E. Gubbins, R. Radhakrishnan, and M. Sliwinska-Bartkowiak, Rep. Prog. Phys. 62, 1573 (1999). C. Rascón and A. O. Parry, Nature (London) 407, 986 (2000). E. H. Hauge, Phys. Rev. A 46, 4994 (1992). D. B. Abraham, A. O. Parry, and A. J. Wood, Europhys. Lett. 60, 106 (2002). C. Rascón and A. O. Parry, Phys. Rev. Lett. 94, 096103 (2005). A. O. Parry and C. Rascón, J. Chem. Phys. 132, 204704 (2010). L. Bruschi, A. Carlin, and G. Mistura, Phys. Rev. Lett. 89, 166101 (2002). G. Gallavotti, Commun. Math. Phys. 27, 103 (1972). D. B. Abraham and P. Reed, Phys. Rev. Lett. 33, 377 (1974). D. B. Abraham and P. Reed, Commun. Math. Phys. 49, 35 (1976). D. B. Abraham, Phys. Rev. Lett. 44, 1165 (1980). D. Abraham, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic Press, London, 1986), Vol. 10, p. 1. D. B. Abraham and F. T. Latremoliere, Phys. Rev. Lett. 77, 171 (1996). M. E. Fisher, J. Stat. Phys. 34, 667 (1984). D. B. Abraham and A. Maciolek, Phys. Rev. Lett. 89, 286101 (2002). D. B. Abraham and A. Maciolek, Phys. Rev. E 72, 031601 (2005). D. B. Abraham, V. Mustonen, and A. J. Wood, Europhys. Lett. 63, 408 (2003). K. Rejmer, S. Dietrich, and M. Napiorkowski, Phys. Rev. E 60, 4027 (1999). A. O. Parry, C. Rascón, and A. J. Wood, Phys. Rev. Lett. 83, 5535 (1999).
†
[8]
[12] [13] [14] [15] [16] [17] [18]
[19] [20] [21] [22] [23] [24] [25]
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[26] A. O. Parry, C. Rascón, and A. J. Wood, Phys. Rev. Lett. 85, 345 (2000). [27] A. Malijevský and A. O. Parry, Phys. Rev. Lett. 110, 166101 (2013). [28] G. Delfino and J. Viti, J. Stat. Mech. (2012) P10009. [29] G. Delfino and A. Squarcini, Ann. Phys. (Amsterdam) 342, 171 (2014). [30] The same matrix acts on space-time coordinates x0 ¼ t and x1 ¼ x.
week ending 8 AUGUST 2014
[31] R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The Analytic S-Matrix (Cambridge University Press, Cambridge, 1966). [32] G. Delfino and A. Squarcini, J. Stat. Mech. (2013) P05010. [33] S. Ghoshal and A. B. Zamolodchikov, Int. J. Mod. Phys. A 09, 3841 (1994). [34] D. B. Abraham and A. Maciolek, Europhys. Lett. 101, 20006 (2013).
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PRL 113, 066101 (2014)
Phase Separation in a Wedge: Exact Results Gesualdo Delfino* and Alessio Squarcini† SISSA—International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy and INFN—Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34127 Trieste, Italy (Received 14 March 2014; published 4 August 2014) The exact theory of phase separation in a two-dimensional wedge is derived from the properties of the order parameter and boundary condition changing operators in field theory. For a shallow wedge we determine the passage probability for an interface with endpoints on the boundary. For generic opening angles we exhibit the fundamental origin of the filling transition condition and of the property known as wedge covariance. DOI: 10.1103/PhysRevLett.113.066101
PACS numbers: 68.08.Bc, 11.10.Kk, 68.35.Rh
Interfacial phenomena at boundaries are a subject of both experimental and theoretical relevance which has received continuous and extensive interest in the last decades [1–4]. An aspect particularly important for applications is that the structure and geometry of the substrate can alter the adsorption characteristics of a fluid in an important way (see [5] for a review). Adsorption measurements can then be used, for example, to characterize fractally rough surfaces [6], or the connectivity of porous substrates [7]. The basic case of a wedge-shaped substrate [8] acquired special interest since phenomenological and thermodynamic arguments [9] indicated a specific relation with the adsorption properties of a completely flat surface: the wedge wetting (or filling) transition occurs at the temperature for which the contact angle of a fluid drop on a flat substrate equals the tilt angle α of the wedge, a circumstance that allows us to regulate the transition temperature adjusting α. The connections between adsorption characteristics for different opening angles are known as properties of wedge covariance [10–12] and are experimentally testable [13]. It is clearly important to pass from a macroscopic to a fundamental statistical mechanical description. In two dimensions the essential role of fluctuations was established by the exact lattice results for the Ising model on the half plane [14–19], which provided essential support for heuristic statistical descriptions of the wetting of a flat boundary [20]. For the wedge geometry the existence of the filling transition was established for the Ising model on a planar lattice forming a right-angle corner [21,22] (see also [23] for a 60° opening angle on the triangular lattice), but otherwise the theoretical investigations in two and three dimensions have been based on effective interfacial Hamiltonian models [10–12,24–26] or density functional methods [27]. In this Letter, we derive general exact results for phase separation in a two-dimensional wedge. This is achieved by exploiting low energy properties of bulk two-dimensional field theory [28,29] together with a characterization of the 0031-9007=14=113(6)=066101(5)
operators responsible for the departure of an interface from a point on a boundary. For a shallow wedge we determine the exact passage probability of an interface with endpoints on the boundary. The theory provides a fundamental meaning to the contact angle and, for generic α, yields the filling transition condition. More generally, wedge covariance is shown to originate from the properties of the boundary condition changing operators in momentum space. We begin the derivation considering a two-dimensional system at a first order transition point, close enough to a second order transition to allow a continuous description in terms of a Euclidean field theory on the plane with coordinates (x, y). We label by an index a ¼ 1; …; n the different coexisting phases, denote by σðx; yÞ the order parameter operator, and by hσia its expectation value in phase a. The Euclidean field theory corresponds to the continuation to imaginary time t ¼ iy of a relativistic quantum theory in one space dimension, for which phase coexistence amounts to the presence of degenerate vacua jΩa i associated to the different phases of the system. The elementary excitations are kink states jK ab ðθÞi interpolating between different vacua jΩa i and jΩb i; the rapidity θ parametrizes the energy and momentum of these particles as p0 ¼ mab cosh θ and p1 ¼ mab sinh θ, respectively, the kink mass mab being inversely proportional to the correlation length. The relativistic dispersion relation ðp0 Þ2 − ðp1 Þ2 ¼ m2ab is preserved by Lorentz boosts M Λ 11 with matrix elements [30] M00 Λ ¼ M Λ ¼ cosh Λ and 01 10 MΛ ¼ MΛ ¼ sinh Λ, which shift rapidities by Λ; as a consequence, relativistic invariant quantities depend on rapidity differences. If H and P are the Hamiltonian and momentum operators of the quantum system, a local operator Φ satisfies Φðx; yÞ ¼ eixPþyH Φð0; 0Þe−ixP−yH . We want to consider the system in the wedge geometry of Fig. 1, where the points (0, R=2) are boundary condition changing points. More precisely, a boundary field points in direction a (respectively, b) in order parameter space for jyj > R=2 (respectively, jyj < R=2)
066101-1
© 2014 American Physical Society
week ending 8 AUGUST 2014
PHYSICAL REVIEW LETTERS
PRL 113, 066101 (2014)
F μα ðθÞ ¼ F μ0 ðθ þ iαÞ:
y α
ð2Þ
The order parameter in the wedge for jyj < R=2 reads R/2
b 0
hσðx; yÞiW aba ¼ α
a
hΩa jμab ð0; R2 Þσðx; yÞμba ð0; − R2 ÞjΩa i−α ; Z W aba ð3Þ
x where jΩa iα ¼ M −iα jΩa i0 and
−R/2
Z W aba ¼ α hΩa jμab ð0; R=2Þμba ð0; −R=2ÞjΩa i−α :
α FIG. 1 (color online). Wedge geometry with boundary condition changing points at (0, R=2) and an interface running between them.
on the wedge, inducing a symmetry breaking in the bulk. If we denote by a subscript W aba the statistical averages in the wedge geometry with these boundary conditions, the limit for x → ∞ of the order parameter hσðx; yÞiW aba is expected to tend to hσia if R is finite, and to hσib if R is infinite. For mab R large this should correspond to an interface running between the points (0, R=2), separating an inner phase b from an outer phase a (Fig. 1), and whose average midpoint distance from the wedge diverges with R. To see how this emerges from the theory, consider first the case of a completely flat boundary (tilt angle α ¼ 0). With a uniform boundary condition of type a on the whole boundary we have a theory defined on the half plane x ≥ 0 with a vacuum state that we denote jΩa i0 . The change of boundary condition at the point (0, R=2) then corresponds to the insertion of an operator μab ð0; R=2Þ responsible for the emission from the boundary of kinks interpolating between phase a and phase b. The simplest matrix element of such a boundary operator is R jK ba ðθÞi0 ¼ e−mðR=2Þ cosh θ F μ0 ðθÞ; 0 hΩa jμab 0; 2
ð1Þ
We expand Eq. (4) by inserting a complete set of asymptotic states between the two boundary condition changing operators, take the limit of large mR in which the lightest (single kink) intermediate state dominates, and obtain Z ∞ dθ μ 2 Z W aba ∼ F α ðθÞF μ−α ðθÞe−mR½1þðθ =2Þ : ð5Þ 0 2π An emission amplitude F μ0 ð0Þ ≠ 0 would mean that the kink has a probability of remaining stuck on the boundary, a possibility that we are excluding for the time being. Hence, on general analyticity grounds [31], we have F μ0 ðθÞ ¼ cθ þ Oðθ2 Þ, so that Eq. (2) gives F μα ðθÞ ∼ cðθ þ iαÞ;
jθj; jαj ≪ 1;
ð6Þ
c2 e−mR ð1 þ mRα2 Þ: Z W aba ∼ pffiffiffiffiffiffi 2 2π ðmRÞ3=2
ð7Þ
and then
In a similar way, we evaluate Eq. (3) expanding over intermediate states and taking the limits mR ≫ 1, m½ðR=2Þ − jyj ≫ 1, which project onto the single kink states. We then obtain hσðx; yÞiW aba ×
where we set mab ≡ m; jK ba ðθÞi0 is an asymptotic kink state (outgoing if θ > 0) on the half plane and has energy E0 þ m cosh θ, E0 being the energy of the vacuum jΩa i0 . If we now move to a new frame through the Lorentz boost M Λ , the kink rapidity becomes θ þ Λ, and the boundary moves in time. For a purely imaginary Λ ¼ −iα the boost corresponds to a rotation in the Euclidean plane, and to a boundary forming an angle α with the y axis. By relativistic invariance the emission amplitude F μα ðθ − iαÞ of the kink with rapidity θ − iα from the moving boundary coincides with F μ0 ðθÞ, something that we can rewrite as
ð4Þ
e−mR ∼ Z W aba
Z
þ∞
−∞
dθ1 dθ2 μ F α ðθ1 ÞMσ ðθ1 jθ2 Þ ð2πÞ2
2 2 F μ−α ðθ2 Þe−ðm=2Þ½ððR=2Þ−yÞθ1 þððR=2ÞþyÞθ2 þimxðθ1 −θ2 Þ ;
Mσ ðθ1 jθ2 Þ ≡ hK ab ðθ1 Þjσð0; 0ÞjK ba ðθ2 Þi;
ð8Þ ð9Þ
evaluating the matrix element, Eq. (9), on bulk kink states we imply that the leading boundary effects for large mR are accounted for by the boundary condition changing operators inserted at the points (0, R=2). In the small rapidity limit which gives the leading contribution to the integral in Eq. (8) the matrix element of the order parameter operator takes the form [28,29]
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Mσ ðθ1 jθ2 Þ ∼ −
iΔhσi þ 2πδðθ1 − θ2 Þhσia ; θ1 − θ 2
ð10Þ
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where Δhσi ≡ hσia − hσib . The delta function term is a disconnected part corresponding to the kink trajectory passing aside the operator without interacting; the pole comes from the fact that the operator sees different phases if the kink passes to its right or to its left. We use Eq. (6) to evaluate Eq. (8) for small positive α, but also differentiate with respect to x in order to get rid of the pole in Eq. (10). The result is pffiffiffi m 3=2 ðx þ Rα ∂ x hσðx; yÞiW aba Þ2 − ðαyÞ2 −χ 2 pffiffiffi 3 2 e ; ∼8 2 R Δhσi π κ ð1 þ mRα2 Þ ð11Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where we defined κ ¼ 1 − 4y2 =R2 , χ ¼ 2m=Rðx=κÞ. Integrating back over x with the asymptotic condition hσðþ∞; yÞiW aba ¼ hσia gives pffiffiffiffiffiffiffiffiffiffi 2 χ þ 2mR ακ −χ 2 hσðx;yÞiW aba ∼ hσib þ Δhσi erfðχÞ − pffiffiffi : e π 1 þ mRα2
FIG. 2 (color online). Passage probability density Pðx; yÞ=m for the interface, with Pðx; yÞ given by Eq. (11), mR ¼ 25 and α ¼ 0.04. The leftmost contour line corresponds to Pðx; yÞ ¼ 0, and then to the wedge. The required divergence of the passage probability density as the pinning points (0, R=2) are approached is also clearly visible.
Z
ð12Þ For α ¼ y ¼ 0 and hσia ¼ −hσib , this result coincides with that originally obtained in [17] from the exact lattice solution of the Ising model on the half plane. It was shown in [28,29,32] that the leading term in the large mR (small rapidity) expansion corresponds to a sharp separation between pure phases, with subsequent terms accounting for the internal interface structure. In the present geometry the order parameter for sharp phase separation can be written as Z hσðx; yÞiW aba ∼ hσia
x~
x
Z dvPðv; yÞ þ hσib
∞
dvPðv; yÞ;
x
ð13Þ where Pðv; yÞ is the probability that the interface passes in the interval (v, v þ dv) on the line of constant ordinate y, which intersects the wedge at x ¼ x~ ðyÞ. It follows that Pðx; yÞ coincides with Eq. (11). R ∞It can be checked that the consistency requirement x~ dxPðx; yÞ ∼ 1 implies pffiffiffiffiffiffiffi mRα ≪ 1. The result, Eq. (11), shows, in particular, that Pðx; yÞ vanishes for jyj ¼ x=α þ R=2, which in the small α approximation of the present computation are the coordinates of the wedge (x ≥ −Rα=2). This shows that the properties, Eqs. (2) and (6), that we identified in momentum space for the matrix element of the boundary condition changing operator indeed lead to an impenetrable wedge in coordinate space. A plot of Pðx; yÞ is shown in Fig. 2. The average midpoint position of the interface reads
x¯ ¼
∞
−Rα=2
dx x Pðx; 0Þ
rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi 2R πmR 2 1þ α þ OðmRα Þ : ¼ πm 2
ð14Þ
The results, Eqs. (11) and (12), apply to values of temperature [i.e., of bulk correlation length m−1 ∝ ðT c − TÞ−ν ] for which the kink state is the lightest one entering the spectral decomposition of Eq. (3). To discuss the situation in which this is not the case we start again from α ¼ 0. For temperatures below a certain threshold T 0 < T c the kink K ba ðθÞ may form with the boundary a bound state jΩ0a i0 with energy E00, in which the phase b forms a thin layer adsorbed on the boundary. As usual for stable bound states, this corresponds to a purely imaginary rapidity θ ¼ iθ0 of the kink, leading to a binding energy E00 − E0 ¼ m cos θ0
ð15Þ
smaller than m. This boundary bound state is now the lightest state contributing to Eq. (3) and produces an order parameter equal to hσia for mx ≫ 1, no matter how large R is [32]. Since m is the interfacial tension between the phases a and b [28], and E0 (respectively, E00 ) corresponds to the tension between the boundary and phase b (respectively, a), Eq. (15) identifies θ0 as the contact angle of the phenomenological wetting theory. The usual relation θ0 ðT 0 Þ ¼ 0 characterizing the wetting transition temperature T 0 corresponds to the unbinding threshold for the kink.
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a
µab
derivation makes transparent the relation of wedge covariance with the relativistic nature of the kink excitations, and explains why this property has been difficult to establish in general within nonrelativistically covariant effective interfacial models. The passage probability for an interface with endpoints on the wedge has also been determined for small tilt angles in the unbound regime.
µab a
b
g b
FIG. 3 (color online). Illustration of Eq. (16). The kink emission amplitude exhibits a bound state pole corresponding to adsorption of phase b on the boundary.
Bound states manifest in matrix elements as poles in the physical region of kinematical variables [31]. For the matrix element, Eq. (1), this physical region corresponds to the strip Im θ ∈ ð0; πÞ, and the pole induced by the boundary bound state takes the form (see Fig. 3) F μ0 ðθÞ ¼ 0 hΩa jμab ð0; 0ÞjK ba ðθÞi0 ig ∼ hΩ jμ ð0; 0ÞjΩ0a i0 ; θ − iθ0 0 a ab
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*
[1] [2]
[3]
[4] [5]
θ ∼ iθ0 ;
ð16Þ
[6] [7]
with g a coupling measuring the strength of the bound state. It then follows from Eq. (2) that the pole of F μα ðθÞ is located at θ ¼ iðθ0 − αÞ. Since the kink energy is always m cosh θ and the unbinding threshold remains at θ ¼ 0, the filling transition temperature T α is determined by the condition
[9] [10]
θ0 ðT α Þ ¼ α:
[11]
ð17Þ
For θ0 < α the pole is located at Im θ < 0, namely, outside the physical strip allowed for bound states; in such a case the kink is unbounded and phase b fills the wedge. The condition, Eq. (17), is that obtained in the macroscopic framework [9], and follows here from the exact statistical theory. Notice that while Eqs. (11) and (12) rely on Eq. (6), and then on small α, Eqs. (2) and (17) are general. Equation (2), in particular, encodes the essence of wedge covariance. For the scaling Ising model on the half plane with a boundary magnetic field h, the scattering amplitude off the boundary is known exactly [33], and exhibits a boundary bound state pole corresponding to 1 − sin θ0 ¼ h2 =ð2mÞ ¼ ðT c − T 0 ðhÞÞ=ðT c − TÞ; the last equality follows from νIsing ¼ 1 and holds in the scaling limit (see, e.g., [34] for boundary bound states in the lattice formalism). In summary, we constructed the exact theory of phase separation in a two-dimensional wedge and derived from it the filling transition condition and the origin of wedge covariance. This has been achieved directly in the continuum and for arbitrary opening angles, while previous exact results concerned only the lattice Ising model and were limited to a right-angle corner. In particular, our
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†
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