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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 901 (2015) 430–443

www.elsevier.com/locate/nuclphysb

Multiple phases and vicious walkers in a wedge

Gesualdo Delﬁno

a,b,∗

, Alessio Squarcini

a,b

SISSA – Via Bonomea 265, 34136 Trieste, Italy

INFN – sezione di Trieste, Italy

Received 16

September 2015; received in revised form 22 October 2015; accepted 28 October 2015

Available

online 3 November 2015

Editor: Hubert

Saleur

Abstract

consider a statistical system in a planar wedge, for values of the bulk parameters corresponding to

a ﬁrst order phase transition and with boundary conditions inducing phase separation. Our previous exact

ﬁeld theoretical solution for the case of a single interface is extended to a class of systems, including the

Blume–Capel model as the simplest representative, allowing for the appearance of an intermediate layer of

a third phase. We show that the interfaces separating the different phases behave as trajectories of vicious

walkers,

and determine their passage probabilities. We also show how the theory leads to a remarkable form

of wedge covariance, i.e. a relation between properties in the wedge and in the half plane, which involves

the appearance of self-Fourier functions.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Fluid interfacial phenomena at boundaries form an important chapter of statistical physics

and are studied experimentally, theoretically and numerically (see [1–10] for reviews). On the

theoretical side, the exact results obtained for the lattice Ising model in two dimensions [11,

12,3] provided an important benchmark for approximated or heuristic approaches, but proved

too dif

ﬁcult to extend to other universality classes. Only recently it has been shown that phase

separation and the interfacial region in planar systems can be described exactly for the different

Corresponding author.

E-mail

addresses: delﬁno@sissa.it (G. Delﬁno), alessio.squarcini@sissa.it (A. Squarcini).

http://dx.doi.org/10.1016/j.nuclphysb.2015.10.019

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

G. Delﬁno, A. Squarcini / Nuclear Physics B 901 (2015) 430–443 431

universality classes [13–15] relying on low energy properties of two-dimensional ﬁeld theory

[16]. This new approach also allowed the exact solution [17] of the longstanding problem of

phase separation in a wedge, which received much attention [18–26] as the basic example of the

effect of the geometry of the substrate on the adsorption properties of a ﬂuid. P

articular interest

was attracted by emergent relations between adsorption properties in the wedge and those on a

ﬂat substrate. Observed at the macroscopic level [18] and successively referred to as properties

of “wedge covariance”, this type of relations resisted a derivation within a statistical mechanical

framework. It wa

s one of the results of [17] to show for the planar case how wedge covariance

follows from the relativistic invariance of the quantum ﬁeld theory associated to the universality

class, which in turn reﬂects the homogeneous and isotropic nature of the ﬂuid.

The wedge problem has been considered so fa

r for the case in which the universality class

and the boundary conditions lead to the separation of two phases a and b. Here we develop

our exact ﬁeld theoretical approach to study the case in which a macroscopic bubble of a third

phase c forms in-between

the two favored by the boundary conditions. Establishing whether a

third phase will intrude between phases a and b forming a macroscopic intermediate layer or

just microscopic droplets at the interface is a main question of wetting physics. It was shown in

[13,14] that in two dimensions the answer is determined by the spectrum of elementary e

xcita-

tions of the underlying ﬁeld theory, which is known for the different universality classes. A model

which leads to the macroscopic (wetting) layer of the third phase and which we will consider in

this paper is the q-state Potts ferromagnet [27] at its ﬁrst order transition point, at which the q

ferromagnetic phases (tw

o of which correspond to phases a and b) coexist with the disordered

phase (which will play the role of phase c). In two dimensions the ﬁrst order transition corre-

sponds to a critical temperature T = T

for q>4 [28]; strictly speaking, our ﬁeld theoretical

description is exact for the scaling limit q → 4

[29], but is expected to remain quantitatively

meaningful up to q ≈ 10, where the correlation length is still much larger than lattice spacing.

Our derivation applies also to q<4 provided we allow for the possibility of vacant sites (dilu-

tion); then coexistence of the disordered with the ordered phases is recovered above a critical

lue of dilution for a value T

of the temperature. For q =2 one obtains a dilute Ising model,

also known as Blume–Capel model, for which the wetting character of the disordered phase has

been investigated numerically [30–32].

We will wo

rk for values of the boundary parameters such that the inner phase is not adsorbed

on the boundary. This means that the third phase is separated from the other two-by-two inter-

faces which ﬂuctuate between the two boundary conditions changing points. We will see how

this picture emerges within the ﬁeld theoretical frame

work and will determine the passage prob-

abilities for the interfaces, ﬁnding in particular that they randomly ﬂuctuate with the constraint

of avoiding each other and the boundary, i.e. that they correspond to trajectories of so-called “vi-

cious” walkers [33]; in this way we determine the passage probabilities of vicious wa

lkers in a

wedge. Concerning the issue of wedge covariance, it turns out to acquire additional interest in the

case of two interfaces, with a surprising interplay between physical considerations in momentum

space and mathematical realization of the condition of impenetrability of the wedge.

The deri

vations are exact and apply to the case of a shallow wedge, that for which the re-

sults are universal, in the sense that they do not depend on the speciﬁc values of the boundary

parameters, as long these are in the range which does not bind the interfaces to the bound-

ary

The paper is or

ganized as follows. In the next section we recall the setting and the results of

[17]. This will put us in the condition of developing the theory for the case of two interfaces in

Section 3. The ﬁnal section is then devoted to summary and comments.

432 G. Delﬁno, A. Squarcini / Nuclear Physics B 901 (2015) 430–443

2. Two phases in a wedge

We start with the characterization of the statistical system in absence of boundaries, i.e. on

the inﬁnite plane. The system is considered at a ﬁrst order phase transition point, where different

phases, that we label by an index a = 1, 2, ..., n, have the same free energy and can coe

xist at

equilibrium. At the same time the system is supposed to be close to a second order transition

point,

in such a way that the correlation length is much larger than microscopic scales and

a continuous description is allowed. For homogeneous and isotropic systems this continuous

description is provided by a Euclidean ﬁeld theory with coordinates (x, y) identifying a point on

the plane. This ﬁeld theory in turn corresponds to the continuation to imaginary time t =iy of

a quantum ﬁeld theory in one space dimension with coordinate x. The de

generate phases of the

statistical system are in one-to-one correspondence with degenerate vacua |0

 of the associated

quantum theory. We denote by σ(x, y) the order parameter ﬁeld, and by σ 

=0

|σ(x, y)|0



the value of the order parameter in phase a. For a generic ﬁeld  we have

(x,y) = e

yH−ixP

(0, 0)e

−yH+ixP

, (1)

with the Hamiltonian H and momentum operator P of the quantum system acting as generators

of time and space translations, respectively; the vacuum states carry zero energy and momentum.

As usual in presence of de

generate vacua in (1 + 1) dimensions (see [16]), the elementary

excitations correspond to kinks |K

(θ) which interpolate between two different vacua |0

 and

, and whose energy and momentum satisfy the relativistic dispersion relation

(e, p) =

(

cosh θ,m

sinh θ

)

, (2)

where m

is the kink mass (inversely proportional to the bulk correlation length) and θ is known

as rapidity. Two vacua |0

 and |0

 (as well as the corresponding phases) are said to be ad-

jacent if they can be connected by an elementary kink; when the connection requires a state

(θ

) ...K

n−1

(θ

), with n necessarily larger than one, the two vacua are said to

be non-adjacent.

As a further step to

wards the study of the wedge problem, we consider this statistical system

on the half plane x  0. We call boundary condition of type a a uniform (i.e. y-independent)

boundary condition at x = 0 favoring phase a in the bulk,

in such a way that the order parameter

approaches σ 

as x →+∞. We will use the notation |0



for the vacuum state of the quantum

system on the half line with this boundary condition; more generally, the subscript 0 will be used

to indicate the presence of the vertical boundary.

Phase separation can be induced through a change of boundary conditions. W

ithin the ﬁeld

theoretical description, the change of boundary conditions from type a to type b at a point y on

the boundary is realized by the insertion of a ﬁeld μ

(0, y), with non-zero matrix elements on

states interpolating between |0



and |0



. When these two vacua are adjacent, which is the

case we consider in this section, the simplest matrix element of μ

0

|μ

(0,y)|K

(θ)

−my cosh θ

(θ) , (3)

As an example, for the Ising ferromagnet these speciﬁcations amount to consider a temperature slightly below the

critical value T

, in absence of external ﬁeld.

In a ferromagnet this is achieved applying a magnetic ﬁeld on the boundary.

Here and below, in order to simplify the notation, we drop the indices on the kink mass.

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