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\documentstyle[12pt]{article}
\title{Stable Quasicrystalline Ground States}
\author{Jacek Mi\c{e}kisz \\ Institut de Physique Th\'{e}orique \\
Universit\'{e} Catholique de
Louvain  \\ 1348 Louvain-la-Neuve, Belgium}
\pagenumbering{arabic}
\begin{document}
\baselineskip=26pt
\maketitle
{\bf Abstract.} We give a strong evidence that noncrystalline materials
such as quasicrystals or
incommensurate solids are not exceptions but rather are generic in some
regions of a phase space. We
show this by constructing classical lattice gas models with
translation-invariant
finite range interactions and with unique quasiperiodic ground states which
are stable
against small perturbations of finite range potentials. \\

\section{Introduction}

One of the important problems in physics is to understand why matter is
crystalline at low
temperatures \cite{and,br,sim,uhl1,uhl2,rad1,rad2}. It is traditionally
assumed (but has never been
proved) that at zero temperature minimization of the free energy of a
system of many interacting
particles can only be obtained by their periodic arrangements (a perfect
crystal) which at nonzero
temperature is disrupted by defects due to entropy. Recently, however,
there has been a growing
evidence, that this basic phenomenon, the crystalline symmetry of low
temperature matter, has
exceptions; in particular incommensurate solids \cite{aub} and, more
recently, quasicrystals
\cite{jarr}. It is very important to find out how generic are these
examples. In other words, is
nonperiodic order present in these systems stable against small
perturbations of interactions between
particles?

The problem of stability of quasiperiodic structures was studied recently
in continuum models of
particles interacting through a well, Lennard-Jones, and other potentials
\cite{wid1,jar1,jar2,jar3,ola,smi}. However, no final conclusion was
reached. After all one has to
compare chosen quasiperiodic structure with all possible arrangements of
particles in the space, a
really formidable task.

Here we will present two classical lattice gas (toy) models with unique
stable nonperiodic ground
states. More precisely, every site of the simple square lattice can be
occupied by one of several
different particles. The particles interact through two-body finite range
translation-invariant potentials. Our models has only nonperiodic ground
state configurations
(infinite lattice configurations minimizing the energy density of the
system). However, they all look
the same: there is a unique translation-invariant ground state measure
supported by them and which is
a zero temperature limit of the grand canonical ensemble. We will prove
that if one perturbs our
models a little bit, by introducing sufficiently small chemical potentials
or two-body
translation-invariant interactions, their ground states do not change. It
means that there is an
open ball in the space of finite range interactions (with a fixed range)
without periodic ground
states. This constitutes a first generic counterexample to the crystal
problem which is an attempt to
deduce, within statistical mechanics, periodic order in systems of many
interacting particles.

In Section 2 we introduce a strict boundary property for local ground
states and explain why it
implies stability of our nonperiodic ground states. In Section 3 we
describe the main features of
Robinson's nonperiodic tilings of the plane, construct a classical lattice gas
 model out of it, and
show why its unique ground state is not stable against small perturbations
of nearest neighbor
potentials. In Section 4 we present a modification of Robinson's tilings
which allows us to construct
models with unique nonperiodic ground states satisfying the strict boundary
property. Section 5
contains a short discussion.

\section{Classical Lattice Gas Models and Nonperiodic Ground States}

A classical lattice gas model is a system in which every site of a lattice
$Z^{d}$ can be
occupied by one of $n$ different particles. An infinite lattice
configuration is an assignment of
particles to lattice sites, that is an element of $\Omega =
\{1,...,n\}^{Z^{d}}$. Particles can
interact through two-body finite range translation-invariant potentials
$f(\underline{x}-\underline{y})$. Configurations of particles minimizing
the potential energy density
of a Hamiltonian
$H=\sum_{(\underline{x},\underline{y})}f(\underline{x}-\underline{y})$ are
called
ground state configurations. For any potential the set of ground state
configurations is nonempty but
it may not contain any periodic configurations
\cite{rad3,rad4,rad5,mier,mie1,mie2,mie3,rad6}. We
restrict ourselves to systems in which, although all ground state
configurations are nonperiodic,
there is a unique translation-invariant measure (called a ground state)
supported by them. The
unique ground state is then inevitably a zero temperature limit of an
infinite volume grand canonical
probability distribution. This is in analogy with the Ising
antiferromagnet, where there are two
alternating ground state configurations but only one ground state measure
which is just their
average. Such ground state configurations have then necessarily uniformly
defined frequencies for all
finite arrangements of particles. More precisely, to find a frequency of a
finite arrangement in a
given configuration we first count the number of times it appears in a box
of size $l$ and centered
at the origin of the lattice, divide it by $l^{d}$, and then take the limit
$l \rightarrow \infty$.
If the convergence is uniform with respect to the position of the boxes
then we say that the
configuration has a uniformly defined frequency of this arrangement. It is
a main point of this paper
to claim that stability of nonperiodic ground states is intimitely
connected with the rate of this
convergence.

{\bf Definition:} Let $\rho$ be a unique ground state measure of a finite
range
potential.  Let $X(A)$ be a configuration on a finite region $A$ of the
lattice
enclosed by a perimeter P and such that all interactions in $A$ attain
their minimal values. $X(A)$ is
then a local ground state configuration but might not be extendable to any
infinite lattice ground
state configuration in the support of $\rho$. We say that a model satisfies
a strict boundary
property for local ground states if for any particle or a nearest neighbor
pair of particles (any
finite arrangement in general) the number of its appearance $n$ in $A$
satisfies the following
inequality: $|n-wA|<CP$, where $w$ is a frequency of the arrangement in
$\rho$ (frequency in any
infinite lattice ground state configuration in the support of $\rho$), and
$C$ is a constant which
depends only on the arrangement.   $\Box$

Now, we claim that the unique ground state of the model satisfying the
strict boundary property for
local ground states is stable in the space of nearest neighbor (finite
range in general)
interactions. For the proof assume otherwise, that a perturbed model has a
different ground state.
This implies the existence of a ground state configuration with a nonzero
density of nearest neighbor
pairs of particles with nonminimal energies of interactions - the so called
broken bonds. The absence
of broken bonds is a complete characterization of the original ground state
as explained in Sec.3.
Now, every broken bond introduces a positive bounded below energy, say $1$.
It is a unit of a
boundary of a local ground state configuration which can decrease energy by
an amount at most
proportional to the strength of a perturbation which is a consequence of the
 strict boundary
property. It follows that for sufficiently small perturbations the energy
of a configuration is
proportional to the total length of broken bonds (a Peierls condition is
satisfied). Hence for
sufficiently small perturbations the hypothetical ground state has an
energy density bigger than the
original one which is a contradiction.


\section{Robinson's Tilings and Unstable Ground States}

To construct our classical lattice gas model we will use Robinson's
nonperiodic tilings of
the plane \cite{rob,gr}. He designed 56 square-like tiles such that using
an infinite number of their
copies one can tile the plane only in a nonperiodic fashion. We describe
now the Robinson tiles; we
follow \cite{rob} closely. There are five basic tiles represented
symbolically in Fig.1. The rest of
them can be obtained by rotations and reflections. The first tile on the
left is called a cross; the
rest are called arms. All tiles are furnished with one of the four parity
markings shown in Fig.2.
Crosses can be combined with the parity marking at the lower left in Fig.2.
Vertical arms (the
direction of long arrows) can be combined with the marking at the lower
right and horizontal
arms with the marking at the upper left. All tiles may be combined with the
remaining marking. Two
tiles ``match'' if arrow heads meet arrow tails separately for the parity
markings and the markings
of crosses and arms.

We now describe main features of Robinson's nonperiodic tilings. Let us
observe that if the plane is tiled with tiles with the above parity
markings then these must alternate
both horizontally and vertically in the manner shown in Fig.2. Now, every
odd-odd position on the
$Z^{2}$ lattice (if columns and rows are suitably numbered) is occupied by
crosses in relative
orientations as in Fig.3. (crosses are denoted there by $\lfloor$ with line
segments corresponding to
double arrows). They form a periodic configuration with period $4$. Then in
the center of each
``square'' matching rules force a cross such that the previous pattern
reproduces but this time with
period $8$. Continuing this procedure (the so called principle of expanding
squares) infinitely many
times we obtain a nonperiodic configuration. It has built in periodic
configurations of period
$2^{n}$, $n\geq 2$ with tiles on sublattices of $Z^{2}$ as shown in Fig.4.

Now using Robinson's tiles we construct a lattice gas model
\cite{rad3,rad4,rad5,mier,mie1,mie2,mie3,rad6}. Every site of the square
lattice can be occupied by
one of the 56 different particles-tiles. Two nearest neighbor particles
which do not ``match''
contribute a positive energy, say $1$; otherwise the energy is zero. Such a
model obviously does not
have periodic ground state configurations. In fact, there is a one-to-one
correspondence between
infinite tilings and configurations in the support of the unique ground
state measure. It has been
proven \cite{mier} that the unique ground state of this model is not stable
against small
perturbations of chemical potentials. Really, let us introduce a small
negative chemical potential
$h$ for arms with a single long arrow (called later single arms). Notice
that every ground state
configuration contains an arbitrarily long sequence of arms with two long
arrows (called later double
arms). If we change such sequence of length $n$ into a sequence of single
arms we lower the energy by
$nh$ paying only at two endpoints of the sequence (the model does not
satisfy the strict boundary
property). So no matter how small is $h$ one can always change a ground
state configuration locally
to lower its energy. This shows, in fact, that densities of single and
double arms change
continuously with respect to their chemical potentials. This continuous
change in stoichiometries at
zero temperature is a behavior not present in any previously known models
and real alloys.

\section{Stable Quasiperiodic Ground States}

We will construct models for which introducing a chemical potential for one
kind of arms (any
two-body interaction in general) does not  change their ground states. Simply
 getting rid of, say,
double arms will not work because these are exactly double arms which
create squares at whose centers
crosses are forced. The absence of double arms will destroy the principle
of expanding squares and we
will no longer have a unique nonperiodic ground state. However, let us
notice that the forcing is
done by four arms located exactly in the middle between crosses forming a
square. Now, the main idea
of our construction is to force similar arms in the middle of every side of
every square on every
scale as in the Robinson original tiling, and to do it not by sequences of
single and double arms but
by sequences of new markings. These will be sequences of repeated $A$, $B$,
and $C$
markings (explained later in detail) propagating from crosses. Whenever two
 $B$ or $C$ markings meet
at the middle of a side of a square then our special arms (described later)
are forced there. Observe,
that we do not have long sequences of different arms anymore, just three
translates of $...ABCABC...$
lines. Obviously, it is impossible to distinguish between the translates by
a chemical potential and
also by two-body interactions because of the "incommensurability" of
$2^{n}$
"periods" of Robinson's tilings and period $3$ of the lines, and the
symmetry of tiles.

MODEL I

Let us now describe tiles of our first model. We have modified crosses and
arms, and we have
the parity markings as before. In addition, we introduce two levels of
letter markings. Each side of
each tile is marked by $A$, $B$, or $C$ and by $T$, $O$, $M$, $E$, or $K$
(not all choices are
present as explained below). All possible markings of crosses are shown in
Fig.5. Let us stress at
this point that we allow all rotations of crosses but we do not allow any
reflections which makes our
model somewhat different from the original Robinson one. Now, we have $15$
types of arrows present in
arms: all combinations of $A \rightarrow B$, $B \rightarrow C$, $C
\rightarrow A$ and $T \rightarrow
O$, $O \rightarrow M$, $M \rightarrow E$, $E \rightarrow K$, $K \rightarrow
T$. Line segments present
in arms are marked $A-A$, $B-B$, or $C-C$; $B-B$ and $C-C$ segments are
also marked $T-O$ or $O-T$ as
shown in Fig.6. Again, we do allow rotations but not reflections. Now, two
tiles match if in addition
to previous requirements letter markings match separately on both levels.
We also assume that line
segments $A-A$ can be put next to any letter marking of the second level
(this is not a tiling
condition). Our matching rules can be translated as before into a lattice
gas model with nearest
neighbor interactions. In addition we have to take care of the fact that
letter markings of the second
level which are opposite nearest neighbors of $A-A$ line segments are not
forced to be correlated. We introduce a two-body interaction of range $2$
between two arrows facing
one another such that the energy is zero if one of them has $T$ marking and
the other one is marked by
$O$; otherwise the energy is $1$.

Now we will show that the only ground state configurations of our model are
those of
Robinsons's type shown in Fig.4. More precisely, looking at crosses and
ignoring
letters and $A$ arrows on every sublattice $2^{n}Z^{2}$ with an odd $n$ and
$C$ arrows for an even $n$
we see the self-similar structure of expanding squares shown in Fig.4.

\newtheorem{metro}{Proposition}
\newtheorem{dodo}{Theorem}

\begin{metro}
The unique ground state measure of Model I is of Robinson's type as
described above.
\end{metro}

{\bf Proof:} First, let us recall that the parity markings force crosses to
occupy every site
of a $2Z^{2}$ sublattice. Letter markings of the second level allow there
only crosses which are
rotations of the cross at the upper left in Fig.5. Now, arms with the upper
left
and lower right parity marking force relative orientations of these crosses
to be such that squares
of length $2$ are created as shown in Fig.3 (letters, arrows with $A$
markings and arms are not
drawn). The above arms with $C-C$ line segments force crosses to appear in
the center of every square
just created. New crosses (four rotations of the cross at the upper right
in Fig.5) form a $4Z^{2}$
sublattice.  Now, the nature of our interactions is such that there are
sequences of arrows
propagating from every cross and meeting half-way through line segments of
arms. For the crosses on a
$2^{n}Z^{2}$ sublattice with an odd $n$ these are alternating $C-C$ and
$A-A$ line segments and
alternating $B-B$ and $A-A$ ones for an even $n$. $C-C (B-B)$ segments,
located in the middle of four
sides of the squares they have created, force crosses in their centers and
together with $A-A$
segments force them to have relative orientations on every sublattice as in
Fig.3. Crosses on
succesive sublattices have to be chosen in the  clockwise order from Fig.5.
The upper left corner of
every square has to be occupied by an appropriate cross from Fig.5 and
other corners by its clockwise
rotations. Inductively an infinite nonperiodic configuration is created.
$\Box$


\begin{metro}
Model I satisfies the strict boundary property for local ground states if
in counting particles and pairs of particles we do not distinguish between
their different
rotations and reflections.
\end{metro}

{\bf Proof:} Let a broken bond be a unit segment on the dual lattice
separating two nearest neighbor
particles with a positive interaction energy (a common side of two nearest
neighbor tiles which do not match). Let us divide $A$ into connected
components without
broken bonds (two lattice sites are connected if they are nearest
neighbors) such that on every
component crosses having  relative orientations  shown in Fig.3 form a
$2Z^{2}$ sublattice. This can
be achieved by paths on the lattice dual to $Z^{2}$ with lengths smaller
than $16$ and joining broken
bonds. When one considers only $2Z^{2}$ sublattices and lines of arms
joining their sites the
deviation of the number of the appearance of a given particle from the
number of its appearance in a
global ground state configuration is not bigger than $16$ times the number
of the above paths. Now,
we decompose further every connected component into connected components
such that crosses form there $4Z^{2}$ sublattices. This time it is achieved
by paths on the dual
lattice with lengths smaller than $32$ and again joining broken bonds. Now,
when one considers
$4Z^{2}$ sublattices and lines of arms joining their sites, the additional
deviation of the occurence
of our particle is not bigger than $16$ times the number of new paths plus
$16/2$ times the number of
old paths. At the $n-th$ step we divide every connected component
constructed so far into connected
components with crosses forming $2^{n}Z^{2}$ sublattices and using paths
with lengths smaller than
$2^{n+3}$. Again, the additional deviation of the occurence of our particle
is bounded by $16$ times
the number of new paths plus $1/2$ of the deviation at the $n-1th$ step. We
repeat this process $m$
times, where $m$ is the smallest number such that $diam A <m$. Now, the
total number of all paths is
bounded by $3$ times the number of broken bonds (broken bonds and paths can
be regarded as vertices
and edges of a planar graph and our bound follows from Euler's formula).
This shows that $|n-wA|
\leq 96P$, where $w$ is a frequency of the particle in a global ground
state and $n$ the number of its
appearance in $X(A)$. Similarly one can obtain the same bound for any
two-particle arrangements.
$\Box$

Let us notice that our model does not satisfy the strict boundary property
for three-particle
arrangements which leads to instability in the space of three-body
interactions. To see this let us
look at a line of arms with the length $k$. $T$ and $O$ markings of arrows
of tiles immediately above
form a nonperiodic sequence with the following property: if a certain
fraction of particles is
ignored the rest of a ground state configuration is periodic; the smaller
the fraction, the larger
the period. When one just looks at two-particle arrangements than the above
proof applies because for
every $T$ above the line there $O$ below it and vice versa, and in counting
 arrangements we do not
distinguish between reflected ones. Let us look now at two consecutive
markings and create a
nonperiodic sequence of two symbols $a$ and $b$, where $a$ corresponds to
$TT$ and $OO$, and $b$ to
$OT$ and $TO$ in the original sequence. Shifting the line  of arms we may
then increase up to $\log
k$ (one for every sublattice $2^{n}Z^{2}$, $2^{n} \leq k$ the number of,
say, $A \rightarrow B$ arrows
with the $a$ symbol immediately above them. The shift creates only two
contours hence it follows
that the strict boundary property is not satisfied.

Our first zero temperature stability result will be stated in the space of
covariant
interactions.  That it is to say that a chemical potential of a
particle-tile or a two-body
interaction between a pair of particles should be the same for any of their
rotations and reflections.
The following theorem follows directly from the strict boundary property as
shown in the introduction.

\begin{dodo}
The unique nonperiodic ground state of Model I is the unique ground state
for any sufficiently
small two-body covariant perturbation.
\end{dodo}

MODEL II

Let us now describe tiles of our second model. We have modified crosses
with $A$ and $C$ arrows
shown in Fig.7, arms with $A \rightarrow B$, $B \rightarrow C$, and $C
\rightarrow A$ arrows and
$A-A$, $B-B$, and $C-C$ line segments shown in Fig.8 and 9. Arms in Fig.7
are combined with the
upper right parity marking and in addition an arm at the bottom left can be
combined with the lower
right or upper left parity marking. Arms in Fig.8 can be combined with the
lower right or upper
left parity marking. Crosses are combined with the parity markings as
before. We allow all
rotations and reflections of our new tiles as in the original Robinson
model.

We do not have anymore the previous second level of letter markings. $B-B$
and $C-C$ line segments,
not located in the middle of arms anymore, still force crosses to create
squares but symmetric $A-A$
arms cannot prevent these squares going out of phase; a fault shown in
Fig.10 can be created. The
presence of such a fault does not produce any broken bonds if and only if
all line segment on it are
of the $A-A$ form. It follows that in any tiling there may be at most one
fault so they have zero
probability of occurence with respect to the unique translation-invariant
measure supported by
tilings. For exactly the same reasons such faults are present in some of
the original Robinson
tilings. Therefore as in the previous model the unique ground state of
Model II is of the Robinson
type. Looking at crosses and ignoring letter markings and $A$ arrows on
every sublattice $2^{n}Z^{2}$
with an odd $n$ and $C$ arrows with an even $n$ we again see the
self-similar structure of expanding
squares shown in Fig.4.

\begin{metro}
Model II satisfies the strict boundary property for local ground states if
in counting particles and pairs of nearest neighbor particles we do not
distinguish between
their different rotations and reflections.
\end{metro}

{\bf Proof:} The first step is the same as in the proof of Proposition 1.
In the second step we
divide further connected components such that on every component crosses
form
$4Z^{2}$ sublattices but not necessarily having relative orientations as in
Fig.3;
faults may be present. We may have an arbitrarily long fault along
perpendicular $A-A$ line segments
without any broken bonds except at the endpoints of the fault. The presence
of
two arms on the fault with line segments parallel to the fault forces a
cross between them and then a
broken bond is forced at a distace smaller than $16$ from the cross as can
be seen from Fig.10. Arms
with $B-B$ or $C-C$ line segments perpendicular to the fault also force a
cross on the fault. It
follows that when one considers only $4Z^{2}$ sublattices and lines of arms
joining their sites the
deviation of the number of the appearance of a given particle from the
number of its appearance in a
global ground state configuration is not bigger than $16$ times the number
of paths in the present
decomposition plus $16/2$ times the number of paths in the first step. (there is
 no change
of numbers
of particles and pairs of nearest neighbors along a fault with $A-A$ line
segments perpendicular to
it). Now, we decompose further every connected component into connected
components such that crosses
form there $8Z^{2}$ sublattices which of course go out of phase along
previous faults and they may
also create they own faults. The induction based on self-similarity of our
tilings can now be applied
to obtain the same bound as before. $\Box$

The following theorem follows.

\begin{dodo}
The unique nonperiodic ground state of Model II is a unique ground state
for any sufficiently
small nearest-neighbor covariant perturbation.
\end{dodo}

Modifying slightly the previous argument involving three-body interactions
it can be shown that the
ground state of the second model is not stable against arbitrarily small
perturbations of range $2$.


Let us notice that particles of our models have shapes and interactions
between them are covariant
but not symmetric: an interaction between nearest neighbors $RP$ is the
same as between
\hspace{5mm} but not necessarily the same as between $PR$. There is however
a construction due to
Radin \cite{rad4} who by enlarging a family of possible particles makes
them shapeless and
interactions between them fully symmetric (dependent only on the distance
between particles). He
constructs in this way a classical lattice gas model with a finite range
fully symmetric interactions
and with unique nonperiodic ground state. His method requires the original
family of particles-tiles
to be both rotation and reflection invariant. Therefore it can be applied
only to our second model.
We use it to construct a model with shapeless particles and with a unique
nonperiodic ground state
which is stable in a space of fully symmetric interactions.

\begin{dodo}
The unique ground state of a symmetric version of Model II is stable in the
space of
symmetric  interactions with the range smaller than three lattice spacings.
\end{dodo}

\section{Conclusions}

We have constructed finite range classical lattice gas models with unique
quasiperiodic ground
states which are stable against small perturbations of interactions.

In the construction of our models we used the presence of order in
nonperiodic ground state
configurations. Although nonperiodic they are not however completely
chaotic. They exhibit
long-range positional order in the sense that states at distant regions are
correlated. In fact, our
ground state configurations possess highly ordered structures: if a certain
fraction of particles is
ignored the rest of a ground state configuration is periodic; the smaller
the fraction, the larger
the period. It would be interesting to see if other, more disordered ground
states could
be stable. Modifying the principle of expanding squares responsible for
nonperiodicity of Robinson's
tilings a large family of models without periodic ground states was
recently constructed
\cite{moz,rad6,tsi}. In another class of models we have ones based on
Penrose tilings but with
particles on the simple square lattice. Low temperature behavior of such
models was recently studied
in \cite{wid2}. Their stability would be especially interesting in
connection with quasicrystals. It
was proven recently \cite{zahr} that Penrose matching rules satisfy the
strict boundary property for
finite regions of perfect infinite tilings. It is not known if this is also
true for local tilings -
ground states.

Finally, let us say few words about positive temperatures, where to
construct stable equilibrium
phases of a system of many interacting particles one has to minimize not
the energy but the free
energy of the  system taking into account disruptions due to entropy. Here
one of the important
problems is to construct a quasiperiodic equilibrium phase for a finite
range interaction - a
microscopic model of a quasicrystal. We conjecture the existence of
quasiperiodic equilibrium
phases in a modified three-dimensional version of our second model.

{\bf Acknowledgment.} This research was supported by Bourse de recherche UCL/FDS

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