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\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}
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\begin{document}
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\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|