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%%% ON THE MICROSCOPIC THEORY OF PHASE COEXISTENCE %%%
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%%% Author: Salvador MIRACLE SOLE %%%
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%%% June 1994 %%%
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{
\centerline{{\twelve Centre de Physique
Th\'eorique}\footnote{$^{\star}$}{\eightrm Unit\'e Propre de Recherche
7061}{\twelve - CNRS - Luminy, Case 907}}
\centerline{{\twelve F--13288 Marseille Cedex 9 -
France}}
\vskip 4truecm
\centerline{{\ten ON THE MICROSCOPIC THEORY}}
\centerline{{\ten OF PHASE
COEXISTENCE}\footnote{$^{\star\star}$}{\eightrm Contribution to the XIII
Sitges Conference on Statistical Mechanics,
Sitges-Barcelona, Spain, June 1994}}
\bigskip
\centerline{ {\bf Salvador Miracle Sol\'e} }
\vskip 2truecm
\centerline{\bf Abstract}
\medskip
Some rigorous results concerning
the microscopic theory of interfaces and crystal shapes
in classical lattice systems
are reported.
\vskip 3truecm
%\noindent Number of figures : 1
\bigskip
\noindent October 1994
\noindent CPT-94/P.3077
\bigskip
\noindent anonymous ftp or gopher : cpt.univ-mrs.fr
\footline={}
\vfill\eject }
\noindent{\pc 1. Introduction}
\medskip
\noindent
It is known that the equilibrium shape of a crystal
is obtained,
according to the Gibbs thermodynamic theory,
by minimizing the total surface free energy
associated to the crystal-medium interface,
and that this shape is given by the Wulff
construction,
provided one knows the anisotropic
surface tension
(or interfacial free energy per unit area).
It is therefore important,
even if a complete microscopic derivation
of the Wulff construction
within statistical mechanics
has been proved only for
some two-dimensional lattice models
(see the recent work by Dobrushin {\it et al.} [1,2]
and also Ref.\thinspace 3),
to study the properties of the surface tension
$\tau ({\bf n})$,
as a function of the unit vector
${\bf n}$
which specifies the orientation of the
interface with respect to the crystal axes.
In the first approximation the crystal can be
modelled by a lattice gas.
In these notes, we shall present some new rigorous results
on this subject which relate,
in particular, to the problem of the
appearance of plane facets in the Wulff
equilibrium shape.
For this purpose several aspects of the microscopic
theory of interfaces are analysed,
and another important quantity in this theory,
the step free energy, is investigated
(a complete version of this work will appear later [4]).
In the last Section we shall report on
some recent developments related to the theory
of crystal growth.
\bigskip
\noindent{\pc 2. Gibbs states and interfaces}
\medskip
\noindent
First we recall some classical results about the Gibbs
states of the
Ising model (to some extent these results were already
discussed in Ref.\thinspace 5).
The model is defined on the cubic lattice
${\cal L}=\relatif^3$,
with configuration space
$\Omega = \{ -1, 1\}^{\cal L}$. The value
$\sigma (i)$ is the spin at the site $i$.
The energy of a configuration
$\sigma_{\Lambda} = \{\sigma(i), i\in \Lambda \}$,
in a finite subset $\Lambda \subset {\cal L} $,
under the boundary conditions ${\bar \sigma}\in\Omega$,
is
$$
H_{\Lambda}(\sigma_{\Lambda}\mid\bar{\sigma})
= - \sum_{\langle i,j \rangle \cap \Lambda \not= \emptyset}
\sigma (i)\sigma (j)
$$
where
$\langle i,j \rangle$ are pairs of nearest neighbour sites
and
$\sigma (i) = \bar{\sigma} (i) $ if
$ i \not\in \Lambda $.
The partition function, at the inverse temperature
$\beta=1/kT$,
is given by
$$
Z^{\bar\sigma}(\Lambda)
=\sum_{\sigma_{\Lambda}}\exp \big(-\beta
H_{\Lambda}(\sigma_{\Lambda}\mid{\bar\sigma})\big)
$$
It is known that this model presents, at low temperatures
$T0$,
we expect this interface,
which at $T=0$ coincides with the plane $i_3=-1/2$,
to be modified by deformations.
It can be described by means of its
defects, or excitations,
with respect to the interface at $T=0$.
These defects, called {\it walls},
form the boundaries
(which may have some width), between the smooth plane
portions
of the interface.
In this way
the interface structure, with its probability distribution
in
the corresponding Gibbs state,
may then be interpreted as
a ``gas of walls'' on
a two-dimensional lattice.
Using the Peierls method, Dobrushin [6] proved
the dilute character of this gas at low temperatures,
which means that the interface is essentially
flat (or rigid).
The considered boundary conditions yield indeed a
non translation invariant Gibbs state.
Furthermore,
cluster expansion techniques have been applied
by Bricmont {\it et al.} [7,8],
to study the interface structure in this case
(see also Ref.\thinspace 9).
The same analysis applied to the two-dimensional model
shows a different behaviour at low temperatures.
In this case the walls belong to a one-dimensional lattice,
and
Gallavotti [10] proved that the microscopic
interface undergoes large fluctuations of order
$\sqrt{L_1}$. The interface does not survive in the
thermodynamic limit, $\Lambda\to\infty$,
and the corresponding Gibbs state is translation invariant.
Moreover, the interface structure can be studied by means
of a cluster expansion for any orientation of the interface
(see also Ref.\thinspace 11).
Such a problem
in the three-dimensional case leads
to very difficult problems of random surfaces.
This is one of the
serious difficulties which face the attempts to generalise
the work by Dobrushin {\it et al.} [1]
to the three-dimensional Ising model,
since a very accurate description of the microscopic
interface for any orientation ${\bf n}$
is needed in this work.
\vfill\eject
\noindent
{\pc 3. The surface tension}
\medskip
\noindent
The free energy, per unit area, due to the presence of the
interface,
is the surface tension. It can be defined by
$$
\tau({\bf n})=
\lim_{L_1,L_{2}\rightarrow\infty} \, \lim_{L_3
\rightarrow\infty} \, -{{n_1} \over {\beta L_1 L_2}}
\ln \, {Z^{(\pm,{\bf n})}(\Lambda)\over
Z^{(+)}(\Lambda)}
$$
Notice that in this expression
the volume contributions
proportional to the free energy of the coexisting phases,
as well as the boundary effects, cancel, and only
the contributions to
the free energy of the interface are left.
\medskip
\noindent
\pc Theorem 1. \it
The thermodynamic limit
$\tau ({\bf n})$, of the interfacial free energy
per unit area, exists, and, as a function of
${\bf n}$, extends by positive homogeneity to
a convex function
$f({\bf x})=\vert {\bf x} \vert \, \tau ({\bf x}/ \vert
{\bf x} \vert) $
defined for any vector ${\bf x}\in \Reel^3$.
\rm
\medskip
A proof of these statements was given in Ref.\thinspace 12
using correlation inequalities
(this being the reason for their general validity).
Moreover, we know
(from Refs.\thinspace 13 and 14 and the convexity
condition) that $\tau ({\bf n})$ is strictly positive for
$T 0$ if $T < T_c^{d=2}$,
was proved in Ref.\thinspace 18,
as well as that
$\tau^{\rm step} = 0$ if $T \ge T_c$.
Since then, however, it appears to be no proof of the fact
that
$T_R < T_c$.
At present one is able to study rigorously the roughening
transition
only for some simplified models of the microscopic
interface.
Thus, Fr\"ohlich and Spencer [19] have proved
this transition for the SOS
(solid-on-solid) model, and
several restricted SOS models,
which are exactly solvable, have also been studied
in this context
(these models
are reviewed in Refs.\thinspace 20 and 21).
In order to define the step free energy we
consider the box $\Lambda$ as above and
and introduce the ({\eightrm step},{\bf m})
boundary conditions,
associated to the unit vectors
$ {\bf m} = (\cos \phi,\sin \phi) \in \Reel^2 $,
by
$$
{\bar \sigma }(i) =
\cases{
1 &if $ i>0 $ or if $ i_3=0 $ and $ i_1m_1+i_2m_2\ge 0
$\cr -1 &otherwise \cr}
$$
Then, the {\it step free energy}, for a step
orthogonal to ${\bf m}$ (such that $m_2\ne0$),
is
$$
\tau ^{\rm step}(\phi ) =
\lim _{L_1 \to \infty }
\lim _{L_2 \to \infty }
\lim _{L_3 \to \infty }
- { {\cos \phi}\over {\beta L_1} } \
\ln \ { {Z^{({\rm step},{\bf m})}(\Lambda )}\over
{Z^{(\pm,{\bf n}_0)}(\Lambda )} }
$$
Clearly, this expression represents the residual free
energy due to the considered step, per unit length.
When considering the configurations under the ({\eightrm
step},{\bf m})
boundary conditions, the step may be viewed as a defect
on the rigid interface.
It is in fact,
a long wall going from one side to the other of the box
$\Lambda$.
A more careful description of it can be obtained as
follows. At $T=0$, the step parallel to the axis
(i. e., for ${\bf m}=(0,1)$)
is a perfectly straight step of height 1.
At a low temperature $T>0$, some deformations appear,
connected by straight portions of height 1.
The step structure, with its probability distribution
in the corresponding Gibbs state,
can then be described as a ``gas'' of these defects
(to be called {\it step-jumps}),
on a one-dimensional lattice.
This description, somehow similar to the description of the
interface of the two-dimensional Ising model used by
Gallavotti [10],
is valid, in fact, for any orientation ${\bf m}$ of the
step.
It can be shown that the gas of step-jumps is a dilute gas
at low temperature
and, as a consequence of this fact,
cluster expansion techniques can be applied in order
to study the step structure.
Actually, the step-jumps are not independent since
the rest of the system produces an effective interaction
between them.
Nevertheless, this interaction can be
treated by applying the low temperature expansion,
in terms of walls,
for the rigid interface, to the regions of the interface
lying at both sides of the step.
{}From this analysis one gets the following result.
\medskip
\noindent
\pc Theorem 3. \it
If the temperature is low enough
(i.e., if $T\le T_0 $
where $T_0 > 0$
is a given constant),
then the step free energy $ \tau^{\rm step} $,
exists in the thermodynamic limit, and
extends by positive homogeneity to a strictly convex
function.
Moreover, it can be expressed in terms of an
analytic
function of $T$, for which
a convergent power series expansion
can be obtained from the above mentioned cluster expansion.
\rm
\medskip
In fact,
$$
\eqalignno{
\tau^{\rm step}({\bf m}) =\ \
2J (|m_1|+|m_2|)
&- (1/\beta) \big( (|m_1|+|m_2|) \ln (|m_1|+|m_2|) \cr
&- |m_1|\ln|m_1| - |m_2|\ln|m_2| \big)
- (1/\beta) \varphi_{\bf m} (\beta) \cr}
$$
where $ \varphi_{\bf m} $
is an analytic function
of $z=e^{-2\beta}$,
for $|z| \le e^{-2\beta_0}$.
The first two terms in this expression,
which represent the main contributions
for $T\to 0$,
come from the ground state of the system
under the considered boundary conditions.
The first term can be recognised as the residual energy
of the step at zero temperature
and, the second term, as $-(1/\beta)$ times
the entropy of this ground state.
The same two terms occur in the surface tension of the
two-dimensional Ising model (see Ref.\thinspace 22).
By considering the lowest energy excitations,
it can be seen that $\varphi_{\bf m}$
is $O(e^{-4\beta})$,
and also, that the first term in which this series
differs from the series associated to
the surface tension of the two-dimensional Ising model,
is $O(e^{-12\beta})$.
\bigskip
\noindent
{\pc 5. Facets in the equilibrium crystal}
\medskip
\noindent
The roughness of an interface
should be apparent
when considering
the shape of the equilibrium crystal
associated with the system.
One knows that
a typical equilibrium crystal at low temperatures
has smooth plane facets
linked by rounded edges and corners.
The area of a particular facet
decreases as the temperature is raised
and the facet finally disappears at a
temperature characteristic of its orientation.
The reader will find information and references
on equilibrium crystals in
the review articles of Refs.\thinspace 20, 21, 23 and 24.
It can be argued,
as discussed below,
that the roughening transition
corresponds to the disappearance of the
facet whose orientation is the same as that of the
considered interface.
The exactly solvable SOS models mentioned above,
for which the function $\tau({\bf n})$
has been computed,
are interesting examples of this behaviour
(this subject has been reviewed in Ref.\thinspace 12,
Chapter VII).
For the three-dimensional Ising model,
Bricmont {\it et al.} [25] have proved
a correlation inequality which establish
$\tau^{\rm step}$
as a lower bound to the
one-sided derivative
$\partial\tau(\theta) / \partial\theta |_{\theta=0^+} $
(here $\tau^{\rm step}=\tau^{\rm step}(0,1)$
and $\tau(\theta)=\tau(0,\sin\theta,\cos\theta)$).
Thus $\tau^{\rm step}>0$ implies a kink in $\tau(\theta)$
at $\theta=0$ and, according to the Wulff
construction, a facet is expected.
In fact, $ \tau^{\rm step} $
should be equal to this derivative.
This is reasonable,
since
the increment in surface tension of an interface
tilted by an angle $\theta$,
with respect to the surface tension of the rigid interface,
can be approximately identified, for $\theta$ small,
with the free energy of a ``gas of steps''
(the density of the steps being proportional to $\theta$).
And, again,
if the interaction between the steps can be neglected,
the free energy of this gas can be approximated
by the sum of the individual free energies of the steps.
As a result of the methods described in Section 3,
it is possible to study
the statistical mechanics of
this ``gas of steps'', and
to derive the following result.
\medskip
\noindent
\pc Theorem 4. \it
For $TJ_2>0$,
in the presence of a very small magnetic field $h>0$,
is studied.
Let $\Lambda$ be a square
box of side $L$, with periodic boundary conditions, and let
$H(\sigma)$ denote the energy of a configuration
$\sigma\in\Omega_\Lambda=\{-1,1\}^\Lambda$.
We suppose that the volume is sufficiently large,
$L>(2J_1/h)^3$.
A discrete time {\it stochastic dynamics} is then
considered for this model.
Namely, the Metropolis dynamics
defined by the following updating rule:
Given a configuration $\sigma$ at time $t$ one first chooses
randomly a site $i\in\Lambda$ with uniform probability
$1/\vert\Lambda\vert$.
Then one flips the spin at site $i$ with probability
$$
\exp ( -\beta \max\{H(\sigma^{(i)}) - H(\sigma)\thinspace,
0\})
$$
where $\sigma^{(i)}(j)=\sigma(j)$, whenever $j\ne i$, and
$\sigma^{(i)}(j)= - \sigma(j)$, for $j=i$.
This dynamics is {\it reversible} with respect
to the Gibbs measure.
The nucleation from a metastable state is studied
for this model in the limit of very low temperatures
($h$ fixed).
It turns out that the critical nucleus, as well as the
configurations
on a typical path to it, {\it differ from the Wulff shape}
of an equilibrium droplet.
The critical droplet is in fact a square of side
$\ell^*=[2J_2/h]+1$ ($[\cdot]$ denotes the integer part),
while the Wulff shape is a rectangle of sides
proportional to $J_1,J_2$
(agreement could be expected, however, in the more
customary region
$T$ fixed, small, and $h\to0$).
A {\it path} of the process is a sequence
$\omega=\sigma_0,\sigma_1,\dots,\sigma_t,\dots$ of
configurations
in $\Omega_\Lambda$.
We suppose that the process
starts at the configuration $\sigma_0=(-)$
(all spins $\sigma_0(i)$ in $\Lambda$ equal to $-1$).
We are interested in the first passage
{}from the configuration $(-)$ to the configuration $(+)$,
which takes place between the moments
$\tau_{(-)}=\max \{t<\tau_{(+)} \mid \sigma_t=(-)\}$
and $\tau_{(+)}=\min \{t \mid \sigma_t=(+)\}$.
The configurations $r(\ell_1,\ell_2)$, which have a
rectangle
of sides $\ell_1,\ell_2$ as unique Peierls contour,
play a particular role in the process.
They correspond to the local minima of the energy
(in the sense that one spin flip increases the energy).
Now,
the probability that starting from a given local
minimum $Q$ the system goes to a neighbouring
local minimum $Q'$,
is determined by the energy barrier
$H(S)-H(Q)$,
where $S$ is any configuration at which the energy
on a path from $Q$ to $Q'$ reaches its maximum,
but with the path chosen to minimalise it.
In other words,
the configurations in $S$ are the {\it local saddle points}
for which the minimax
$$
\min_{\omega: Q\to Q'} \max_{\sigma\in\omega} H(\sigma)
$$
is attained
(here $\omega: Q\to Q'$ denotes a generic path with
successive spin flips starting from a configuration in
$Q$ and ending at $Q'$).
The considered probability is proportional to
$\exp[-\beta(H(S)-H(Q))]$.
On the other side,
the system in the local minimum $Q$ is likely to ``stay''
in its basin of attraction for a time of order
$\exp[\beta(H({\bar S})-H(Q))]$,
where ${\bar S}$ is the local saddle point
with lowest energy
through which it can escape from the local minimum
$Q$, not necessarily in ``the direction'' of $Q'$.
These are the basic mechanisms which determine the local
dynamics.
The task is then to find
the class of paths which describe
the most probable evolution.
Let us consider the probability of reaching a
{\it global saddle point},
defined by the same minimax condition extended
to all paths from the configuration $(-)$
to the configuration $(+)$.
These configurations
give rise to the {\it critical nucleus}.
It can be seen that the set of all global saddle points
coincides with the set $\cal P$ of all configurations
having as unique contour a rectangle,
of sides $\ell^*,\ell^*-1$, or $\ell^*-1,\ell^*$,
with a unit square attached to one of its longer sides.
The relative energy of any ${\bar\sigma}\in{\cal P}$
is
$$
\Gamma = H({\bar\sigma})-H((-))=2(J_1+J_2)\ell^*
- h ((\ell^*)^2 - \ell^* + 1)
$$
It is proved that the first excursion from $(-)$
to $(+)$ passes through a configuration from ${\cal P}$
and the time needed for this to happen is of the
order $\exp(\beta\Gamma)$.
Introducing the time
$\tau_{\cal P}=\min\{t>\tau_{(-)} \mid \sigma_t\in
{\cal P}\}$,
the precise statement can be formulated as follows.
\medskip
\noindent
\pc Theorem 5. \it
We have
$$
\lim_{T\to 0} {\rm Prob}\ [\tau_{\cal P}<\tau_{(+)}] = 1
$$
and, moreover, for any $\epsilon>0$,
$$
\lim_{T\to 0} {\rm Prob}\ [\exp(\beta(\Gamma-\epsilon)) <
\tau_{\cal P} < \exp(\beta(\Gamma+\epsilon))] = 1
$$
\rm
In addition, from the arguments in the proof of this result,
one is getting very detailed information about a
typical path followed by the process $\sigma_t$
during its first excursion from $(-)$ to $(+)$.
\medskip
\itemitem{\it a)} {\it
First it passes through a monotonously growing sequence of
subcritical rectangles
$r(\ell_1,\ell_2)$, such that $\vert\ell_1 -\ell_2\vert =
0$ or $1$,
up to the critical square $r(\ell^*,\ell^*)$.}
\itemitem{\it b)} {\it
After the vertical edge stays constant at the value
$\ell^*$
while the horizontal edge grows up to $L$.}
{\it
Finally the vertical edge grows from $\ell^*$ to $L$.}
\medskip
The precise statements involve the notion of
$\epsilon$-typical
path, that is determined not only in terms of geometrical
properties,
but also with specified times of passage
(by means of bounds analogous to those used above for
$\tau_{\cal P}$)
through certain configurations.
The path is an $\epsilon$-typical path (for any given
$\epsilon>0$) with a probability which tends to 1 when
$T\to 0$.
Finally, let us mention that Schonmann [32] has recently
discussed the regime in which the temperature is kept fixed
and the field $h>0$ is scaled to zero.
As conjectured in Ref.\thinspace 33,
for the Ising model in any dimension $d\ge 2$,
if the temperature is low enough,
the relaxation time goes in this regime
as an exponential of $1/h^{d-1}$.
Moreover, before a time which grows also
as an exponential of $1/h^{d-1}$
the system stays in a metastable situation.
\bigskip
\noindent
{\pc Acknowledgements}:
It is a pleasure to thank Roman Koteck\'y for very valuable
discussions.
\bigskip
\noindent
{\pc References}
\medskip
\parindent=6mm
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\end