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\begin{center}
{\bf LOW TEMPERATURE PHASE DIAGRAM FOR A CLASS OF FINITE RANGE INTERACTIONS ON
THE PENROSE LATTICE}\\
\vskip5mm
{\sc F. Koukiou,}\footnote{CPTh, Ecole Polytechnique, F-91128 Palaiseau
Cedex, France, pthflk@frpoly11.earn}
{\sc D. Petritis,}\footnote{IRMAR, Universit\'e de Rennes~I, F-35042 Rennes
Cedex, France, petritis@cicb.fr}
{\sc and M. Zahradn\'\i k}\footnote{KMA, Charles University,
Sokolovsk\'a 83, CS-18600 Praha, Czechoslovakia, umzmz@csearn.earn}
\end{center}
\noindent
Recently, new alloys have been synthesized that exhibit X-ray diffraction
patterns with unusual crystallographic symmetries, excluded by classical
crystallography, like five- or ten-fold axes. It is shown that such patterns
can be obtained by diffraction on quasiperiodic structures like the
Penrose tiling of the plane.
To describe the quasiperiodic lattice, we use the projection method
introduced in \cite{dk}. Namely, we decompose the space $\bR^\nu$ into two
orthogonal subspaces $E_{\|}$ and $E_{\bot}$ and denote by $\pi_{\|}$ and
$\pi_{\bot}$ the corresponding projections. The quasiperiodic lattice is
identified to a particular discrete subset of $E_{\|}$ that will be
constructed in the sequel.
Denote by $\{\epsilon_1,\ldots ,\epsilon_\nu\}$ an orthonormal basis of
$\bR^\nu$, by $\gamma$ the unit hypercube
\[\gamma=\{\xi\in \bR^\nu: \xi=\sum\limits_{i=1}^\nu\xi_i \epsilon_i\ \
\mbox{with}\ \ \xi_i \in [0,1[\},\]
and by $K$ the projection of $\gamma$ on $E_{\bot}$, {\em i.e.}
$K=\pi_{\bot}(\gamma)$. Sliding $\gamma$ over $E_{\|}$ creates a strip
$S=E_{\|}+\gamma$. The projection of the points of the lattice
$\bZ^\nu$ that are intersected by $S$ onto $E_{\|}$ gives the quasiperiodic
lattice we are looking for. Generically, {\em i.e.} for almost
every orientation of $E_{\|}$, the projection $\pi_{\|}$ is bijective
between $\bZ^\nu$ and the $\bZ$-module $\pi_{\|}(\bZ^\nu)$. Therefore,
generically, the quasiperiodic lattice $L$ can be described by
\[L=\{s\in\pi_{\|}(\bZ^\nu): \pi_{\bot}(\pi_{\|}^{-1}(s))\in K\}.\]
For the lattice $L$ to have unusual diffraction
patterns it is necessary $\nu$ to be large enough. Thus, for Penrose
tilings of the plane $\nu=5$, $\dim(E_{\|})=2$ and for tilings of the
three-dimensional space with rhombohedra and diffraction patterns with
eikosahedral symmetry $\nu=6$ and $\dim(E_{\|})=3$.
We turn now to the statistical mechanics on $L$.
Assume that a discrete space $S$ is given and denote by
$X$ the set of all configurations
$x: L\mapsto S$ that will be called {\it configuration space}. For every
{\it finite}
subset $A$ of $L$, denote by $\cF_A=\cS^A$ --- $\cS$ being the exhaustive
$\sigma$-field of $S$ --- and by $\Phi_A$ a $\cF_A$-
measurable map $\Phi_A:X \mapsto \bR$. The $\cF_A$-mesurability condition
implies that $\Phi_A$ depends only on the values of the configuration $x$
over the set $A$. Such functions $\Phi_A$ are called {\it interaction
potentials}.
Assume moreover that $\Phi_A\equiv 0$ whenever $\diam A>r$, where $r$
is some fixed number. Statistical mechanics is as usual described in terms
of the finite volume Hamiltonians
\[H_\Lambda(x)=\sum\limits_A\Phi_A(x)\]
the sum extending over all finite subsets of $L$ that intersect $\Lambda$.
In a previous work, the case of quasiperiodic interaction potentials over
configurations of a regular lattice was considered and the Pirogov-Sinai
theory was proved valid \cite{kpz1}.
Here, the complementary point of view is examined:
the lattice $L$ being quasiperiodic, it is the underlying
configuration space that becomes quasiperiodic. The main result obtained
in \cite{kpz2} can be sketched as follows:
\noindent{\bf Theorem~:~}\em
Assume that
\begin{enumerate}
\item
the Hamiltonian admits a finite number of local ground state configurations,
\item
the interaction potentials are invariant under the
transformations of the symmetry group of $\pi_{\|}(\bZ^\nu)$
\item
the Peierls condition is verified
\item
the temperature is sufficiently low
\end{enumerate}
then, the Pirogov-Sinai theory is applicable.\em
Among the conditions of the theorem, condition 2 needs some clarification.
Let $g$ be an element of the discrete group acting on $E_{\|}$.
Denote by $gA$ the image of the set $A$ under the transformation $g$ and
by $gx_A$ the coresponding image of the configuration $x$ over $A$.
Then, invariance under $g$ means $\Phi_{gA}(gx_A)=\Phi_A(x_A)$.
The following example provides a simple nontrivial case.
\noindent{\bf Example~:~}
Let $L$ be the $2$-dimensional Penrose lattice tiled with two kinds of
rhombs of unit edge and let $a$ and $b$ be the two shortest lengths
appearing in the structure, namely $a=2\sin(\pi/10)$, $b=1$. Let $J_1$, $J_2$
be two positive constants and choose as one spin space the set
$\{-1,1\}$. Then
\[\Phi_A(x)=\left\{\begin{array}{ll}
-J_1x_sx_t&\mbox{if}\ \ A=\{s,t\}\ \ \mbox{with}\ \ \|s-t\|=a\\
-J_2x_sx_t&\mbox{if}\ \ A=\{s,t\}\ \ \mbox{with}\ \ \|s-t\|=b\\
0 &\mbox{otherwise},
\end{array}\right.\]
with $\|\cdot\|$ being the $L_2$ distance in $E_{\|}$, defines a
a variant of Ising models for which our theorem applies.
Other different and more general situations, including many body
interactions, ferromagnetic and antiferromagnetic couplings etc., can
be handled by the present extension.
\begin{thebibliography}{99}
\bibitem{dk} Katz, A., Duneau, M.: Quasiperiodic patterns and eikosahedral
symmetry, {\em J. de Physique} {\bf 47} 181--196 (1986).
\bibitem{kpz1} Koukiou, F., Petritis, D., Zahradn\'\i k, M.: Extension of
the Pirogov-Sinai theory to a class of quasiperiodic interactions,
{\em Commun. Math. Phys.} {\bf 118}, 365--383 (1988).
\bibitem{kpz2} Koukiou, F., Petritis, D., Zahradn\'\i k, M.: Extension of
the Pirogov-Sinai II: the case of quasiperiodic lattice, preprint (1991).
\end{thebibliography}
\end{document}