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%% A SPIN-ONE LATTICE MODEL OF MICROEMULSIONS AT LOW TEMPERATURES %%
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%% by: R. Kotecky, L. Laanait, A. Messager, S. Miracle-Sole %%
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%% To recieve the Figures please send your postal address %%
%% to miracle@cptvax.in2p3.fr %%
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\bigskip
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\centerline{\bf A SPIN-ONE LATTICE MODEL OF MICROEMULSIONS}
\medskip
\centerline{\bf AT LOW TEMPERATURES}
\bigskip
\bigskip
\centerline{\ab Roman Koteck\'y, Lahoussine Laanait,}
\centerline{\ab Alain Messager, Salvador Miracle-Sol\'e}
\bigskip
\centerline{Charles University, Praha, Ecole Normale Sup\'erieure, Rabat,}
\centerline{Centre de Physique Th\'eorique CNRS, Marseille}
\bigskip
\bigskip
{\leftskip=12pt \rightskip=12pt
\noindent
{\ab Abstract:} The phase diagram of a three state microemulsion model
at low temperatures is discussed.
It is shown how, taking into account low energy excitations, the ground
state
phase diagram is modified and the degeneracy of a coexistence line bordering
the region of lamellar phase is taken away.
\par}
\bigskip
{\leftskip=12pt \rightskip=12pt
\noindent
{\ab Key Words:} microemulsion, Gompper-Schick model.
\par}
{\leftskip=12pt \rightskip=12pt
\noindent CPT-93/P.2707
\par}
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\centerline{\ab 1. Introduction}
\medskip
\noindent
Three component mixtures of water, oil and amphiphiles exhibit
very interesting behaviour as temperature and concentration
of the surfactant are varied.
For a microscopic theory of these systems
several hamiltonians have been proposed [1-6].
In a first model, proposed by Widom [1, 2],
formulated in terms of Ising variables, the three species of
molecules occupy, with some constraints, the bonds of a lattice.
More recently, a three component lattice model has been
proposed by Schick, Shih and Gompper [3, 4]
in order to illuminate some additional aspects of amphiphilic systems,
among which are the microemulsion phase and the interfacial properties.
Previous studies [3, 4] based on a mean field theory
present an essential first step
towards understanding the behavior of the model.
However, some features, as the infinite degeneracy of the zero temperature
state
manifold occurring for some values of the amphiphile strength
have not been covered by the mean field approach.
In this note we wish to show how one can account for fluctuations
beyond the mean field
and to study rigorously,
whenever it is possible, the phase diagram of the model.
Namely, we use the standard low temperature perturbation [7]
in order to understand the influence of local excitations and to study
the phase diagram.
The considered model is a simple three component lattice
system in dimension $d\geq 3$.
To each site of a cubic lattice ${\relatif}^d$ is assigned
a spin one variable $s_i$
so that the values $s_i=1,-1,0$
correspond to the presence at site $i$ of a molecule
of water, oil, or amphiphile, respectively.
The hamiltonian
is
$${\Cal H}=\sum_{(i,j)}
J(s_i-s_j)^2 +\sum_{i}(Bs^2_i-Hs_i)+\sum_{(i,j,k)}Ls_i(1-s^2_j)s_k.
$$
The first term is the sum of pair interactions,
assumed to be attractive
between particles of the same kind ($J>0$),
and the second one contains the usual chemical potentials.
With these
two terms the hamiltonian is a particular case of the
Blume-Emery-Griffiths model \cite{8},
which describes a simple three component mixture for the case of
nearest neighbour interactions.
Even though we assume that
two particles of different kind have the same pair interactions,
the discussion
for general pair interactions
can be done by following the same method which
is used in the present work.
The external field $H$ is related to the chemical potential
difference between oil and
water
and $B$ is related to the
chemical potential of the
surfactant.
The third
sum extends over the sets $(i,j,k)$ of three adjacent sites
in a line and $L>0$ is the strength of the amphiphilic interaction.
This term
distinguishes the molecule associated to spin
$0$ as an
amphiphile and mimics its effect by favouring the configuration with
$0$ placed between
$+$ and $-$, all in a line.
\bigskip
\centerline{\ab 2. Phase diagram with low energy excitations}
\medskip
The {\it ground states}
of the model are described as follows. If we assume that
$L=0$, the spins must be equal everywhere since $J>0$. The system has
three ground states, the $(0)$, $(+)$ and $(-)$ states,
where respectively $s_i$ is
equal to $0$, $+1$ and $-1$
for all $i$. The energies per site in these
states are
$$h^{(0)}=0\quad ,\quad h^{(+)}=B-H\quad ,\quad h^{(-)}=B+H$$
and
the corresponding phase diagram is easily obtained.
The
amphiphile interaction favorizes the configuration $(+0-)$ on three
aligned adjacent
sites $(i,j,k)$.
Such configurations appear in ground states
if
$$
L-2J>0
$$
and, for this reason,
we assume hereafter that this condition is satisfied.
Then, at least for some values of $B$ and $H$, it
is clear that the system would favour to have the $L$-bonds (or the sets
of three adjacent sites) occupied as much as possible by such $(+0-)$
configurations. This leads to a new family of ground states with a lamellar
structure.
To describe them more precisely we introduce some notations.
We consider the planes
$r(i)=\alpha _1 i_1+\dots +\alpha_di_d=z$, where all $\alpha$'s
are $+1$ or $-1$ and $z$ is an integer. These planes will be called
{\it diagonal planes},
they may have $2^{d-1}$ possible orientations. We consider the sequence
of all diagonal planes of the lattice with a given fixed orientation. The
configurations that assume a constant value in each plane of such a sequence
will be called {\it layered} configurations.
The {\it lamellar} ground states are the layered
configurations obtained by putting all spins equal to 0 on every second
plane of
the sequence and alternatively equal to $+1$ and to $-1$ on the remaining
planes. In other words,
the sequence of spin values on the diagonal planes is the periodic
sequence $(0+0-\dots)$ of period 4.
This leads to $4.2^{d-1}=2^{d+1}$
equivalent
(i.e., related by translations and rotations of the lattice)
lamellar ground states.
Their energy per site
is
$$
h^{\text{(lam)}}=-{d\over 2}(L-2J)+{1\over 2}B.
$$
By
comparing this expression with those corresponding to the other ground
states
we obtain the following grond state phase diagram (see Fig. 1).
The lamellar states are the unique ground states,
for values of
$B$ and $H$ in the triangular
region
$${B\over d}L-2J+{B\over d}$$
Outside this region
we have the $(+)$ state, if $ H \ge 0 $ and $ H \ge B $,
the $(-)$ state, if $ H \le 0 $ and $ H \le B $ and
the $(0)$ state, if $ |H| \le B $.
We next examine the coexistence lines
of the phase diagram. On the separation line
between the (0) and (lam) states,
only these two states are ground states.
The same
situation occurs on the coexistence lines $(0)/(+)$, $(0)/(-)$ and
$(+)/(-)$.
However, on the coexistence lines $(+)/\text{(lam)}$ and $(-)/\text{(lam)}$
infinitely many
ground states occur.
All of them have a layered structure and may be described by the
sequence of spin values associated to the sequence of parallel diagonal
planes.
A particular role will be played by the periodic ground states
associated to the periodic sequences
$ (0 + ... + 0 - ... -) $,
in which $m \ge 1$ diagonal planes,
where the spins are
$+$, are separated from $n \ge 1$ diagonal planes,
where the spins are
$-$, by single planes, where the spins are $0$.
We use the notation ($m,n$) for these states of period $p=m+n+2$.
The periodic
state (1,1) is the $\text{(lam)}$ ground state considered above.
The energies per site of these states are
$$
h^{(m,n)}
= h^{(+)}-{{2d}\over p}\ (L-2J) - {2\over p}\ B
+{2(n+1)\over p}\ H.
$$
This shows that
all ground states with $(m,n=1)$,
and $(m=1,n)$, are present,
respectively,
on the coexistence lines $(+)/\text{(lam)}$
and $(-)/\text{(lam)}$.
All these states, for any $m$ and $n$,
are present at the triple point,
where these two lines and the $(+)/(-)$
coexistence line intersect.
Our aim is to show that the considered model can be,
in the region $L-2J>0$, rigorously analyzed at low temperatures.
The phase diagram at low temperatures follows from
a competition between energies of ground states taking into
account additional contributions of entropies of low energy excitations.
These ideas are formalized in the
Pirogov-Sinai theory of low temperature phase diagram \cite{7}.
Here we will use an extension of this theory
due to Bricmont and Slawny \cite{9}.
This extension concerns a mechanism of supression of high energy fluctuations.
We first introduce
some notions needed for the study of
equilibrium states at low temperatures.
By an {\it excitation} we understand a configuration which
coincides with a ground state outside of a finite set of sites.
If, in a given
system, the energy of an excitation (relative to the corresponding ground
state) tends to infinity with the size of the region where it differs from the
surrounding ground state, this system is said to be
{\it regular}\footnote*{\eightrm Notice, however, that an excitation may
consists of a region where the configuration is a ground state,
but different from that surrounding the region from outside.}.
The Peierls condition, required
in the Pirogov-Sina\"{\i} theory is satisfied in a given system,
if the energy of an excitation is proportional to
the size of the boundary of this region (or larger).
Clearly, all models in which the
Peierls condition holds are regular.
The method of Bricmont and
Slawny applies to a more general class of regular systems than those satisfying
the Peierls condition.
We shall show that this is the case
for the model under consideration
which is regular but has an infinite number of ground
states and fails to satisfy the Peierls condition.
The two dimensional version of the model is
not even regular, the
lack of regularity which occurs also
in the one-dimensional Ising model.
It is useful to consider the excitations as partial or local
configurations on the lattice, consisting of the set of excited sites, i.e.
those which contribute to the increase of the energy with respect to
that of the
ground state, and an appropriate boundary around this set.
The precise definitions are as follows.
We define a {\it partial configuration}
$X$ by specifying a finite set of sites, called
the {\it domain} of $X$, $\hbox{dom} (X)$,
together with a configuration on this set. We
define a subset of $\hbox{dom} (X)$
called the {\it boundary} of $X$, $\p(X)$, in such a
way that if the configuration on $\p(X)$ is kept fixed, the sites in
$\hbox{dom}(X)\setminus\p(X)$
do not interact with the sites outside of $\hbox{dom} (X)$.
A standard way is to define $\p(X)$ as an $\ell$-boundary, with
$\ell\geq 2$,
i.e., as the set of sites in $\hbox{dom} (X)$
at distance less than $\ell$ from its
complementary set.
A partial configuration $X$ is an {\it excitation}
if its restriction to $\p(X)$ is
a ground configuration on $\p(X)$.
We next present some simple examples assuming only one excited site in the
ground state. We add to this excited site a boundary
consisting of its nearest neighbours and also the next nearest neighbours
on the
same line when the excited site interacts (by an $L$-bond) with them.
We get, for instance the following partial configurations
(actually there are, up to translations and rotations
of the lattice, fourteen
possibilities for such one site excitations).
$$
\matrix{
& & & & & & & & & & & &-& & & & & &+& & \cr
& &+& & & & &-& & & & &0& & & & & &0& & \cr
-&0&0&+ & & +&0&0&- & & -&0&0&0&- & & +&0&0&0&+ \cr
& &0& & & & &0& & & & &0& & & & & &0& & \cr
& &-& & & & &+& & & & &-& & & & & &+& & \cr
& & & & & & & & & & & & & & & & & & & & \cr
& &X_1&& & & &X_2& & & & &X_3&& & & & &X_4&& \cr
}$$
Although only two
dimensions are shown here,
our intention is to represent $d$-dimensional objects
that can easily be understood from the figures,
taking into account the layered structure of ground configurations.
We observe that
all these excitations have the property
of being removable:
an excitation $X$ is {\it removable} if there exists
a unique ground configuration on $\text{dom} (X) $
denoted $G(X)$, whose restriction to $\p (X)$
is equal to the restriction of $X$.
If $X$ is a removable excitation, then its energy $E(X)$, relative to the
ground state, is well defined.
It is the energy of the configuration $X$ minus the energy of $G(X)$.
If $X$ is a removable excitation and $Y$ a partial configuration whose domain
contains $\text {dom}(X)$ and such that its restriction to
$\text {dom}(X)$ is equal to $X$, then there is a unique partial configuration
obtained by removing $X$ from $Y$, equal to $G(X)$ on $\text {dom}(X)$
and to $Y$ otherwise.
The ground configurations
$G_k = G(X_k)$, $k=1,...,4$,
associated to the excitations considered above, are the following ones
$$
\matrix{
& & & & & & & & & & & &-& & & & & &+& & \cr
& &+& & & & &-& & & & &0& & & & & &0& & \cr
-&0&+&+ & & +&0&-&- & & -&0&+&0&- & & +&0&-&0&+ \cr
& &0& & & & &0& & & & &0& & & & & &0& & \cr
& &-& & & & &+& & & & &-& & & & & &+& & \cr
& & & & & & & & & & & & & & & & & & & & \cr
& &G_1&& & & &G_2&& & & &G_3&& & & & &G_4&& \cr
}$$
and the energies $E_k = E(X_k)$, $k= 1,...,4$, of the excitations
are
$$
\matrix{
E_1=-B+H+dL, \hfill &E_2=-B-H+dL, \hfill \cr
E_3=-B+H+2dL-2dJ, \hfill &E_4=-B-H+2dL-2dJ. \hfill \cr
}$$
Since $dL - 2dJ > 0$, we have $E_1 < E_3$ and $ E_2 < E_4 $.
Moreover,
when we consider the system in the vicinity of the $(+)/\text{(lam)}$
or the $(-)/\text{(lam)}$ coexistence lines of the phase diagram of
the ground states, only the
elementary excitations $X_1$,...,$X_4$,
have energy less or equal
than $E = \max \{ E_3, E_4 \}$.
This occurs at least for some range of values of
the coupling constants (namely, if $L < (8/3)J$ )
to which, for concreteness, we shall restrict our
discussion.
If $X$ is a configuration of the system equal to a ground configuration
$G$ outside $\Lambda$, where $\Lambda$ is a large box on the lattice,
then there is a uniquely defined configuration called the
{\it retouch of} $X$,
$ \hbox{ret} (X) $,
obtained from $X$ by removing all elementary excitations with
energy smaller than a given value $E_0$
(i.e., in the case considered above, $E_0=E$, all excitations of types
$X_1$,...,$X_4$). Let ${\Cal E}( G, E_0)$
be the set of such configurations $X$
which, moreover, satisfy
the condition
$ \hbox{ret} (X)=G$.
Namely, it is the set consisting of the ground configuration $G$ and all its
excitations whose energy locally does not exceed $E_0$. We assign to every
configuration in ${\Cal E}( G, E_0)$
the corresponding Boltzmann weight and supress the remaining ones by assigning
them the zero probability. In this way we obtain a state of the system (in the
box $\Lambda $), which will be called the {\it restricted ensemble} associated
to the ground state $G$ (with excitations of energy less than $E_0$).
The partition function
$Z_{\Lambda}^{G,E_0}$,
restricted to the configurations of the set
${\Cal E}( G, E_0)$,
yields the free energy per site
associated to this restricted ensemble
$f_{\Lambda}^{G,E_0} = - (1 / \beta |\Lambda|) \ln Z_{\Lambda}^{G,E_0}$
(in this formula $\beta=1/kT$ denotes the inverse temperature
and $|\Lambda|$ is the volume of $\Lambda$).
This notion extends to the infinite system by
taking the limit $\Lambda$ tending to infinity.
The excitations $Y$, contributing to a restricted ensemble, have
weights $\varphi(Y)=\exp (-\beta E(Y))$.
Two excitations are compatible if their
domains are disjoint. These facts imply that
$$ Z_{\Lambda}^{G,E_0} =
\exp(-\beta{\cal H}(G))
\sum_{\{ Y_1,...,Y_k\} } \varphi(Y_1)...\varphi(Y_k) $$
where ${\cal H}(G)$ is the energy of the ground state $G$
and the sum runs over all sets
$\{ Y_1,...,Y_k\} $, $(k=0,1,2,...)$ of compatible excitations
of $G$ with energy less then $E_0$.
Therefore,
the restricted ensemble can
equivalently be described as a polymer system,
where the polymers are
the different excitations
with the activities $\varphi (Y)$.
Since the system is regular
and $E(Y)\le E_0$,
the number of kinds of these polymers,
up to translations,
is finite (independent of $\Lambda$).
The activities $\varphi (Y)$
are small when the temperature is low.
We use the convergent
small activity expansion (in terms of the Ursell functions)
to compute the free energy of the restricted ensemble.
Since different ground states may have different excitations,
and also different numbers of the common excitations, the free
energies of some of the associated restricted ensembles will be different.
Let us consider the case
$E_0=E=\max\{E_3,E_4\}$ in which all the excitations excitations
to be taked into account are
of types $X_1,...,X_4$ described above.
We use $f^{(m,n),E}$, or simply $f^{(m,n)}$, to denote
the corresponding free energy per site when $G=(m,n)$
is one of the periodic ground states of the system.
In this case,
up to terms of order less than $\exp (-\beta E)$,
the low temperature
expansion can be limited to the first term for each considered excitation,
that is
$$ f^{(m,n)} = h^{(m,n)}
- \lim_{\Lambda \to \infty } (1 / \beta |\Lambda|) \sum_{Y} \varphi(Y) $$
where $h^{(m,n)}$ is the ground state energy per site
and the sum runs over the considered four types of excitations,
contained in $\Lambda$ and
contributing to the restricted ensemble ${\cal E}(G,E)$
of the ground state $G=(m,n)$.
Taking into account the
geometric structure of these grounds states, described above,
we get
$$\eqalign{
f^{(m,n)}
= &\ \ h^{(+)}-{{2d}\over p}\ (L-2J) - {2\over p}\ B
+{2(n+1)\over p}\ H - \cr
&-{2\over p}\ I(m\geq 2)\varphi_1-{2\over p}\ I(n\geq 2)\varphi_2
-{1\over p}\ I(m=1)\varphi_3-{1\over p}\ I(n=1)\varphi_4. \cr
}$$
In
this formula, the first four terms correspond to the energy per site
of the ground state,
$\varphi_{\ell}= (1/\beta) \exp (-\beta E_{\ell})$
for $\ell=1,...,4 $, and $I$ is the indicator of the
condition shown in parentheses (it equals one if the condition is satisfied
and is zero otherwise).
On the other hand, for $G$ equal to the $(+)$ and $(-)$
states, we have
$ f^{(+)}=h^{(+)} $ and $ f^{(-)}=h^{(-)} $
(there are no excitations of these ground states of energies lower than $E$).
Having
the free energies, we may draw the phase diagram for the restricted
ensembles. We say that a point in the $(B,H)$-plane belongs to the $G$-state
region whenever
$$t^{G,E}(B,H) = f^{G,E}(B,H) - \min _G\ f^{G,E}(B,H) = 0 $$
This phase diagram
is schematically represented in Fig. 2.
The region $PQRR'Q'P$ around
the point $B=-d(L-2J)$, $H=0$, is
occupied by the restricted ensemble (2,2). For $H \ge 0$
this region is defined by the
inequalities
$$\eqalignno{
H-{1\over 3} \ (dL-2dJ+B)
&\leq {1\over 3}\ \varphi_2 +{1\over 3}\ \varphi_1,\cr
H+{1\over 3} \ (dL-2dJ+B)
&\leq {5\over 3}\ \varphi_2 -{1\over 3}\ \varphi_1-\varphi_4, \cr
dL-2dJ+B \
&\leq 2\varphi_2 +2\varphi_1-{3\over 2}\ \varphi_4. \cr}$$
The restricted ensemble (2,1) is present in the triangular region
$QRS$, limited by the curve $QR$ already described, and the two curves
$QS$ and $RS$,
where
$$
-2\varphi_1+{1\over 4}\varphi_4\leq H-{1\over 2}(dL-2dJ+B)
\leq {2\over 5}\varphi_1+{1\over 5}\varphi_4.
$$
Since the two
bounds are inconsistent for low temperatures and $E_4 \beta_0$, there exists, in the plane $(H,B)$, an open region
$\Omega(\beta)$ in the complement of the curves $PQS$ and $PQ'S'$ of
the phase
diagram of
the restricted
ensembles (Fig. 2),
whose distance from these curves is of order less than
$\exp(- \beta E)$.
In $\Omega(\beta)$ we have a complete phase diagram of the pure thermodynamic
phases (extremal periodic Gibbs states of the system) which is a small
deformation of the diagram of Fig. 2.
There are in $\Omega(\beta)$, six disjoint open regions, such that
their closure jointly covers $\Omega(\beta)$, that correspond to the
regions denoted $(+)$,$(-)$ and $(m,n)$ with $m\leq 2$ \ and \
$n \leq 2 $, in Fig. 2.
In each of these regions there is a pure phase which is a small deformation
of the associated ground state described above.
The boundaries of these regions are smooth open archs in which two
distinct phases coexist.
They meet, inside $\Omega (\beta )$,
in two points (which correspond to the points
$R$ and $R'$ of Fig. 2) in which three distinct pure phases
coexist.
These coexistence curves are deformations of the
corresponding curves on Fig. 2
of order $\exp(- \beta E)$.
Moreover,
the analysis developped in the present paper
can be pursued by considering subsequent excitations of higher
energies.
This allows to remove degeneracy from the line $PQS$ and
obtain
a rigorous full description of the phase diagram, in a
corresponding region, of
the system at low temperatures.
This is in fact the main result of our work
that we are going to report only
briefly
in the following paragraphs
(a more detailed discussion will be the
subject of a separate publication).
First, we prove
that, with appropiate definitions of the associated
domains and boundaries,
and for any $E$,
all elementary excitaions with energy less than
$E$ are removable.
This allows us to consider
restricted ensembles in which
all these excitations
are taken into account.
We may then
compute their free energy
and,
by using the function
$t^{G,E}(B,H)$
as explained above,
draw the corresponding phase diagram of the
restricted ensembles.
On the other hand,
we prove that,
for any given integer $k$, there is a value $E=E_k$
(which increases linearly with $k$),
such that if
all the elementary excitations with
energy less than $E_k$ are considered,
then a phase diagram may be drawn
that distinguishes all the regions belonging
to the restricted ensembles
associated to the ground states
$(m,n)$ for all $m$ and $n$ such that
$\max \{m,n\} \leq k$.
Then,
as in the case of the lowest energy excitations
that we have already discussed with some detail,
these results lead to a rigorous statement
on the equilibrium states of the system
at low temperatures.
Namely, one can show that
there exists $\beta_0 = \beta_0(k) $
(where $\beta_0(k) \to \infty$ when $k \to \infty$)
such that for all temperatures $\beta > \beta_0$,
there exists in the plane $(H,B)$ an open region $\Omega(\beta)$,
in which we have a complete phase diagram of the pure thermodynamic
phases, with separated regions for
for all pure phases $(m,n)$ for which
$\max \{m,n\} \leq k$.
This phase diagram is a small
deformation of the diagram of the associated restricted ensembles.
A sketch of the phase diagram is shown in Fig. 3.
One finds that the phases are ordered according
to the increasing values of
$(m+n)/p$
when $B$ increases along lines parallel to the $B$ axis
while, going along lines parallel to the $H$ axis,
the values of
$(m-n)/p$
increase when $H$ increases.
A first order phase transition takes place
when crossing the lines of coexistence which
correspond to a discontinuity of one
of the order parameters
$\langle s_{i}^2 \rangle$
or
$\langle s_{i} \rangle$.
Notice that, as we mentioned above,
when $k$ becomes large,
the inverse temperature $\beta _0$,
needed to ensure the existence of the state
$(m,n)$ as a separate phase,
also becomes larger.
Some similarities
between the behaviour of the model considered here and
the behaviour of
the axial next-nearest-neighbour Ising (ANNNI) model
should be clear
(see Refs. 10, 11).
However, the phase diagram that we were discussing above,
is generated in the plane of the
surfactant, oil and water chemical
potentials, with all interactions fixed.
It yields a double infinite sequence of
pure phases indexed by the values of $m$ and $n$.
We notice that a rigorous analysis of the model
proposed by Widom [1,2] has been recently worked out
by Dinaburg and Mazel [12] along lines similar
to those developped in our work.
They found, in the region that they were able to study,
no ANNNI-like behaviour in that model,
contrarily to what was surmised
in some earlier works
(quoted in Ref. 12).
In this case, from infinitely many ground states,
only a small number of pure
phases persists at non-zero temperatures.
The surface tension
behaves quite differently
in systems with a finite and with an infinite number
of ground states.
In our case we find that
the surface tension goes exponentially to zero
as $\beta $ tends to infinity.
This is easy to understand:
an interface between two phases does not cost any
energy, but it costs the free energy of low-energy
excitations.
This is, actually, the basic mechanism that justifies the method
used in this paper.
Since the free energy of these excitations
is exponentially small at low temperatures,
the same is true for the surface tension.
Finaly, the low surface tension between the oil-rich
and water-rich phases,
at a given temperature,
reflects also the low free
energy of the corresponding interface.
In fact,
since the excitations that distinguish
between the ground states $(m,n)$ and $(m-1,n)$
have energy proportional to
$k= \max \{ m,n\}$,
their free energy
decays exponentially when $k$ becomes large.
Therefore,
the surface tension between the phases
$(m,n)$ and $(m-1,n)$
is exponentially small for large $k$.
\bigskip
\bigskip
\centerline{\ab References}
\bigskip
\baselineskip = 12 pt
\item{1.}
B. Widom:
J. Chem. Phys.
{\bf 84},
6943
(1986).
\smallskip
\item{2.}
J.C. Wheeler and B. Widom:
J. Am. Soc.
{\bf 90},
3064
(1968).
\smallskip
\item{3.}
M. Schick and W.H. Shih:
Phys. Rev. B
{\bf 34},
1797
(1986);
Phys. Rev. Lett.
{\bf 59},
1205
(1987).
\smallskip
\item{4.}
G. Gomper and M. Schick:
Phys. Rev. Lett.
{\bf 62},
1647
(1989);
Chem. Phys. Lett.
{\bf 163},
475
(1989);
Phys. Rev. B
{\bf 41},
9148
(1990).
\smallskip
\item{5.}
S. Alexander:
J. Phys. (Paris) Lett.
{\bf 39},
L1
(1978).
\smallskip
\item{6.}
M.W. Matsen and D.E. Sullivan:
Phys. Rev. A
{\bf 41},
2021
(1990).
\smallskip
\item{7.}
S.A. Pirogov and Ya. G. Sina\"{\i}:
Teor. Mat. Phys.
{\bf 25},
1185
(1975);
{\bf 26},
39
(1976).
\smallskip
\item{8.}
M. Blume, V. Emery and R.B. Griffiths:
Phys. Rev. A
{\bf 4},
1071
(1971).
\smallskip
\item{9.}
J. Bricmont and J. Slawny:
J. Stat. Phys.
{\bf 54},
89
(1989).
\smallskip
\item{10.}
M.E. Fisher and W. Selke:
Phil. Trans. R. Soc.
{\bf 302},
1
(1981).
\smallskip
\item{11.}
E.L. Dinaburg and Ya. G. Sina\"{\i}:
Commun. Math. Phys.
{\bf 98},
119
(1985).
\smallskip
\item{12.}
E.L. Dinaburg and A.E. Mazel:
Commun. Math. Phys.
{\bf 125},
27
(1989).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %%
%% ADDRESS %%
%% %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\bigskip
Roman Koteck\'y
\noindent Departement of Theoretical Physics, Charles University,
\noindent V Holevsovick\'ach 2, 180 00 Praha 8, Czechoslovakia
and
\noindent Center for Theoretical Study, Charles University,
\noindent Ovocn\'y trh 3, 116 36 Praha 1, Czechoslovakia
\medskip
Lahoussine Laanait,
\noindent Ecole Normale Sup\'erieure,
\noindent Takaddoum, Rabat, Marocco
\medskip
Alain Messager,
\noindent Centre de Physique Th\'eorique, CNRS,
\noindent Luminy, Case 907, 13288 Marseille Cedex 9, France
\medskip
Salvador Miracle Sol\'e,
\noindent Centre de Physique Th\'eorique, CNRS,
\noindent Luminy Case 907, 13288 Marseille Cedex 9, France
\end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %%
%% FIGURE CAPTIONS %%
%% %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vfill\eject
\nopagenumbers
\parindent 2 truecm
FIGURE CAPTIONS
\bigskip
\item{Fig. 1 --}
The ground state phase diagram
\item{Fig. 2 --}
The sketch of the phase diagram with contributions of the lowest
energy excitations taken into account
\item{Fig. 3 --}
The sketch of a part of the phase diagram at low temperatures
\end
{\it E-mail address:} kotecky@cspuni12.bitnet
{\it E-mail address:} messager@cptvax.in2p3.fr
{\it E-mail address:} miracle@cptvax.in2p3.fr
\end