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\title{Low Temperature Stability of Nonperiodic Structures}
\author{Jacek Mi\c{e}kisz \\ Institut de Physique Th\'{e}orique \\
Universit\'{e} Catholique de Louvain \\ Chemin du Cyclotron, 2 \\ B-1348
Louvain-la-Neuve, Belgium.}
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\begin{document}
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\maketitle
{\bf Abstract.} Complex ground states appearing in classical lattice gas
models and their behaviour in the presence of thermal motions are
discussed.
Our examples consist of a nearest-neighbour model based on nonperiodic
tilings
of the plane and an Ising type model with a fractal ground state.
\eject
One of the important goals of nonlinear dynamics is to discuss complex
structures arising from simple deterministic rules. Here we are concerned
with
an equilibrium or static version of this problem. Namely, the main question
is
as follows: How complicated structures can one get as a result of standard
minimization procedures of equilibrium statistical mechanics? Our models
consist of systems of many particles interacting by short-range
deterministic translation-invariant potentials. The structures we are
interested in are ground states minimizing the energy density of a
system or equilibrium phases minimizing its free energy density.
To be more precise, we will consider here classical lattice gas models in
which
every site of a lattice can be occupied by one of several different
particles.
Particles interact through two-body (many-body) translation-invariant
potentials $f(x-y)$. Configurations of particles on an infinite lattice
minimizing the potential energy density of a Hamiltonian $H=
\sum_{(x,y)}f(x-y)$ are called ground state configurations. It is an old
and
still unsolved problem, the so called crystal problem, to deduce
periodicity
of low temperature matter, i.e., periodicity of ground state configurations
of systems of many interacting particles \cite{rev1,rev2}. It is a converse
of
the crystal problem which is discussed here: How complicated ground state
configurations can short-range, deterministic (as
opposed to random interactions appearing for example in spin glasses)
translation-invariant interactions produce? 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. This is in analogy with the Ising
antiferromagnet, where there are two checkerboard ground state
configurations
but only one ground state measure which is just their average. The unique
ground
state is then inevitably a zero temperature limit of an infinite-volume
grand
canonical probability distribution. The uniqueness of a ground state is
also a
signature of nondegeneracy and genericity of a system.
One of our models is based on Robinson's tiles \cite{rob,pat}. There is a
family
of 56 square-like tiles such that using an infinite number of copies of
each of
them one can tile the plane only in a nonperiodic fashion. This can be
translated into a lattice gas model in the following way first introduced
by
Radin \cite{rad1,rad2,ram,miep}. Every site of the square lattice can be
occupied
by one of the 56 different particles-tiles. Two nearest neighbour particles
which do not "match" contribute positive energy; otherwise the energy is
zero.
Such a nearest-neighbour translation-invariant lattice gas model obviously
does
not have any periodic ground state configurations. In fact, there is a
one-to-one
correspondence between its ground state configurations and Robinson's
nonperiodic tilings. The model has a unique ground state which is highly ordered
(arbitrarily distant regions are correlated). Let us emphasize that the
nonperiodic global order is forced here by local
translation-invariant interactions.
In another class of models we have constructed a one-dimensional Ising spin
model \cite{fra} with $\sigma_{i}=\pm$ at each lattice site $i$ and a
translation-invariant four-body exponentially decaying
interaction:
$$H=\sum_{i=-\infty}^{\infty}\sum_{p=0}^{\infty}\sum_{r=0}^{\infty}exp[-(r+p
^{2}]
(\sigma_{i}+\sigma_{i+2^{r}})^{2}(\sigma_{i+(2p+1)2^{r}}+\sigma_{i+(2p+2)2^{
}})^{2}$$
A unique ground state of this model has two unusual properties: it
has a continuous spectrum and perfect fractal symmetry. The ground state is
supported by nonperiodic Thue-Morse sequences. A Thue-Morse sequence can be
constructed by a repetitive application of the following substitution
rules:
$1 \rightarrow 1 \; -1$ and $ -1 \rightarrow -1 \; 1$
At positive temperatures, in the presence of particles thermal motions (an
analog
of a noise in dynamical systems), to construct stable equilibrium phases of
a
system of many interacting particles one has to minimize not the energy but
the free energy of a system taking into account disruptions due to
entropy.The main question here is: Will complex structure present in ground
states be preserved at positive temperatures or it will be smeared out
by chaotic motions?
Low temperature behaviour of the above described nearest neighbour model
was
investigated in \cite{mie1,mie2,mie3}. The existence of a decreasing
sequence of
temperatures $T_{n}$ was proved, such that if $T