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%\title{\vspace*{-2.4cm}
\title{\vspace*{-2.4cm} Non-periodic long-range order\break
for one-dimensional pair interactions}
\author{
\\
{\normalsize Aernout C. D. van Enter} \\[-1mm]
{\normalsize\it Institute for Theoretical Physics} \\[-1.5mm]
{\normalsize\it Rijksuniversiteit Groningen} \\[-1.5mm]
{\normalsize\it P.O. Box 800} \\[-1.5mm]
{\normalsize\it 9747 AG Groningen} \\[-1.5mm]
{\normalsize\it The Netherlands} \\[-1mm]
{\normalsize\tt AENTER@TH.RUG.NL} \\[-1mm]
%
\\ [-1mm] \and
%
{\normalsize Boguslaw Zegarlinski} \\[-1mm]
{\normalsize\it Department of Mathematics} \\[-1.5mm]
{\normalsize\it Imperial College} \\[-1.5mm]
{\normalsize\it London} \\[-1.5mm]
{\normalsize\it United Kingdom} \\[-1.5mm]
{\normalsize\tt boz@ic.ac.uk} \\[-1mm]
{\protect\makebox[5in]{\quad}}
\\[-2mm]
}
%\date{June 7, 1994}
\date{\datetitle}
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\begin{abstract}
We show how in one dimension non-periodic (quasiperiodic) long-range order
can arise due to long-range translation-invariant pair interactions
with quasiperiodically alternating signs. We discuss the Parisi overlap
distribution between the uncountably many pure states.
%{\bf Key words}: non-periodic long-range order,
%uncountably many pure Gibbs states, Parisi overlap distribution,
%oscillating long-range interactions, Israel-Bishop-Phelps theorem.
\end{abstract}
%\title{\vspace*{-2.4cm}
\clearpage
\setcounter{page}{1}
%\tableofcontents
\section{Introduction}
Since the discovery of quasicrystals \cite{SheBCG} the description of
systems which are ordered in a non-periodic manner, either quasiperiodically
or even more irregularly, has been a topic of considerable interest and
physical relevance, see for example \cite{houche,hof,mie,sen} and the
references mentioned there. Moreover, the study of such irregular types of
long-range order may be of use in understanding some conceptual aspects of
the spin-glass problem.
One question of interest is to show the possibility that such a non-periodic
long-range order arises for spin models with translation-invariant
interactions, in which the spins can take a finite number of values. At zero
temperature this has been achieved for models of
spatial dimension at least two, mostly on a lattice, with nearest neighbour
interactions, based on the theory of non-periodic tilings. In one-dimensional
models, one needs infinite-range interactions \cite{radschu}, which however
can decay arbitrarily fast and can be chosen to be of finite-body type
\cite{garradentmie}. At positive temperatures up to
now there have been only limited results. In one dimension to obtain any type
of long-range order, the interactions are necessarily of infinite range, and,
moreover, they have to decay rather slowly. In \cite{entmie} it was shown that
for some long-range four-body interaction, of a rather artificial form,
long-range order of Thue-Morse type can occur. A related argument for pair
interactions with huge gaps (that is, couplings are non-zero only for pairs of
spins at exceptional distances from each other) was also indicated.
Here we use
similar ideas to show the possibility of {\it a quasiperiodic long-range order
occurring for a translation-invariant pair interaction of alternating sign},
in such a way that the sign of the correlation function essentially follows
the sign of the interaction. In fact, our argument is in some sense inverted,
%goes in the opposite direction,
in that the sign of the wished-for interaction follows the sign
of some quasiperiodic correlation function.
Long-range translation-invariant pair interactions with alternating
sign are physically much more realistic than the interactions considered in
\cite{entmie}. One of the best-known examples is the RKKY-interaction which
plays an important role in the modelling of spin-glasses. For a recent
review on results for long-range interactions we refer to \cite{zeg94}. We
remark that a system with non-periodic long-range order has uncountably
infinitely many pure states, a property which often is suggested to be
characteristic for spin-glasses. (For a further discussion about analogies
between non-periodic systems and spin-glasses, we refer to
\cite{EHM,ent96,zeg94}.) The main difference seems to be the absence of
disorder; however, there have also been attempts to describe
spin-glass behaviour with non-random models \cite{Eis,MarParRit1,MarParRit2,
BM,CKPR, Rit}.
As a side remark we observe that it is much simpler to obtain a non-periodic
(helical) long-range order for continuous vector spin models (compare
\cite{rue}, p.163, note 26). One can
consider, for example, a rotator model with Hamiltonian
\begin{equation}
H = -\Sigma_{i,j} J(i-j){\em cos}(\theta_i - \theta_j - (i-j) \times \alpha)
\end{equation}
for $\alpha$ an irrational multiple of $2\pi$, which is equivalent by a
simple change of variables to a ferromagnetic model if the $J(i-j)$ are
positive. The ferromagnetic long-range order of the transformed model is
equivalent to a non-periodic long-range order in the original model.
\section{The result}
In \cite{zeg94} it was conjectured that under reasonable conditions an Ising
pair interaction whose Fourier transform has its maximum at an irrational point
should show non-periodic long-range order. It can easily be seen, as is also
noticed there, that arguments implying that the behaviour of the Fourier
transform of the interaction function governs the nature of the phase
transition cannot be valid in complete generality.
Here we show some results which at least go in the right direction towards
%obtaining such a result.
confirming this picture.
We consider the circle $[0,2\pi)$ and the irrational rotation map (over an
irrational angle $\alpha $):
\begin{equation}
\ T_{\alpha} \theta =\theta + \alpha
\end{equation}
To any $\theta$ we associate the sequence ${\bf \sigma_{\alpha} ( \theta )}$
$\in$ $\{-1, +1\}^ {Z}$ defined by $\sigma_{n, \alpha} (\theta) = 1$ if $%
\theta + n \alpha$ (mod $2 \pi$) $\in$ $[{\frac{-\pi }{{2}}}, {\frac{+\pi }{{%
2}}})$ and $\sigma_{n, \alpha}(\theta) = -1$ if $\theta + n \alpha$ (mod $2
\pi$) $\in$ $[{\frac{\pi }{{2}}}, {\frac{3 \pi }{{2}}})$. Here $\alpha$ is
an irrational multiple of $2 \pi$.
Then the irrational rotation corresponds to the translation map on these
sequences, and the rotation invariant Lebesgue measure on the circle
corresponds to a translation-invariant, ergodic measure $\mu_{\alpha}$ which
is induced on the space $\{-1, +1\}^{Z}$ by the map $\theta \rightarrow {\bf %
\sigma_{\alpha}(\theta)}$. It is straightforward to compute the pair
correlation functions of these measures, and to see that they have a
non-periodic but quasiperiodic long-range order. Indeed, let
\begin{equation}
\mu _\alpha (\sigma _0\sigma _n)=f_\alpha (n)
\end{equation}
with
\begin{equation}
f_\alpha (n)=1-|{\frac{{((n\times \alpha )\ (mod\ 2\pi ))}}\pi }|
\end{equation}
then the $f_\alpha (n)$ form a quasiperiodic sequence of pair correlation
functions. (The Fourier transform of these pair correlations lives on a
dense set of points in the interval $[0,2\pi )$, given by the integer
multiples of $\alpha $ % plus integer multiples of $2 \pi$
(mod $2\pi $)). It holds that $sgn(f_\alpha (n))=sgn({\em cos}n\alpha )$.
We will apply the Israel-Bishop-Phelps theorem \cite{bishphe, isr1}, \cite
{isr2} Ch.V, \cite{geo88} Ch.16, to conclude the existence of a similar
long-range order for an appropriate Gibbs measure on the configuration-space
(space of sequences) $\{-1,+1\}^{Z}$. For the theory of Gibbs measures for
lattice models, we refer to \cite{EFS,geo88,isr2}. Applying the
Israel-Bishop-Phelps theorem works in the same way in any dimension, for
simplicity we thus consider here the one-dimensional case only. Let us
consider in the following cones of translation-invariant interactions given
by:
\begin{equation}
C_{\alpha} = \{ J(n): sgn (J(n))= sgn (f_{\alpha} (n)), \ \Sigma_n |J(n)|
\leq \infty\}
\end{equation}
such that one has formal Ising Hamiltonians of the form
\begin{equation}
H = - \Sigma_{i,j \in Z} J(i-j) \sigma_{i} \sigma_{j}
\end{equation}
The Israel-Bishop-Phelps theorem in our case reads:
\begin{theorem}
\label{tIBP} For every irrational $\alpha $ and every $\epsilon \geq 0$
there exists a translation-invariant pair interaction $J_{\epsilon ,\alpha
}\in C_\alpha $, such that $\Sigma _n|J_{\epsilon ,\alpha }(n)|\leq {\frac{{%
ln\ 2}}{{\epsilon }}}$, for which there exists a translation-invariant Gibbs
measure $\mu_{J_{\epsilon ,\alpha }}$ which satisfies
\begin{equation}
sgn(\mu _\alpha (\sigma _0\sigma _n))[\mu _{J_{\epsilon ,\alpha }}(\sigma
_0\sigma _n)-\mu _\alpha (\sigma _0\sigma _n)]\geq \ -\epsilon
\end{equation}
\end{theorem}
This means that there exists a Gibbs measure with the same kind of
quasiperiodic long-range order in the two-point correlation functions as
occurs in $\mu_{\alpha}$, while the sign of the interaction between spins at
distance n coincides with the sign of the correlation function at distance
n, at least when this correlation function is not too close to zero. For
pair interactions this puts a limit on the unavoidable frustration.
{\it The decomposition into extremal Gibbs measures of any such a $\mu_{J_{\epsilon ,
\alpha}}$ contains uncountably many elements} (for any extremal non-periodic
Gibbs measure occurring in the decomposition its countably many translates
occur with the same weight, which thus has to be zero; as countably many
elements of measure zero still only add up to a measure zero contribution,
one needs to have uncountably many extremal elements in the decomposition).
To get some more control over the cone of interactions, which gets closer to
the Fourier transform having a maximum at $\alpha$, at the cost of losing
some control over the correlation functions, we can replace $C_{\alpha}$ by its
subcone
\begin{equation}
C^{\prime}_{\alpha} = \{ J(n): sgn(J(n)) = sgn(f_{\alpha}(n)), \ \Sigma_n {%
\frac{J(n) }{{\em cos}n \alpha}} \leq \infty \}
\end{equation}
This means that in the Fourier series $\Sigma_n \ J(n){\em cos}nx$, at the
point $x = \alpha$ not only are all the terms positive, but the $J(n)$ have to
be small whenever the term ${\em %
cos}n \alpha $ is small.
For the corresponding Gibbs measure for some element of the subcone we
obtain the inequality
\begin{equation}
{\frac 1{{\em cos}n\alpha }}%sgn(\mu _\alpha (\sigma _0\sigma _n))
[\mu
_{J_{\epsilon ,\alpha }}(\sigma _0\sigma _n)-\mu _\alpha (\sigma _0\sigma
_n)]\geq -\epsilon
\end{equation}
which still implies that non-periodic long-range order occurs.
{\em Remark}: Similarly to the case of the Fibonacci system \cite{EHM}, it
is possible to compute the Parisi overlap distribution $p(q)$ of the $%
\mu_{\alpha}$. Let us recall that the overlap between two configurations
${\bf \sigma^{1}}$ and ${\bf \sigma^{2}}$ is given by
\begin{equation}
\ q = lim_{N \rightarrow \infty} \ {\frac{1 }{N}} \Sigma_{n=1,...N} \
\sigma^{1}_{n} \sigma^{2}_{n}
\end{equation}
Its distribution with respect to the product-measure $\mu_{\alpha} \otimes
\mu_{\alpha}$ is the Parisi overlap distribution which plays an important
role in the mean-field theory of spin-glasses \cite{mepavi}. In our case it is
given by
\begin{equation}
p(q) \ dq = {\frac{1 }{2}} 1_{[-1,+1]} dq
\end{equation}
which, as one sees, is non-trivial, continuous, and is the same for each
$\alpha$.
This behaviour for the overlap distribution, as well as the earlier results
of \cite{EHM}, illustrates the point emphasised in \cite{NS1,NS2,NS3},
that (also non-trivial) overlap-distributions in great generality are
self-averaging. Here the self-averaging property is trivially fulfilled--there
is no averaging to be done--but the overlap distribution can be explicitly
computed and shown to be non-trivial.
\section{Comments and Conclusions}
We used the Israel-Bishop-Phelps theorem to show the occurrence of
quasiperiodic long-range order at positive temperatures for
translation-invariant pair interactions of (quasiperiodically) alternating
signs.
As usual the problem with applying the Israel-Bishop-Phelps theorem is
its non-constructive character. One knows that there exists an interaction
within a certain class of interactions whose Gibbs states give rise to some
prescribed type of long-range order, but one does not know this interaction
precisely. However, in cases where there is no reflection positivity
available, and also no contour argument has been found, this is essentially
the only rigorous method available to us. It should be helpful for a further
progress in this domain to see that
%at least in principle a
{\it the phase transition with non-periodic long-range order
(with uncountably many
pure Gibbs phases) can occur for a certain type of interaction}.
As compared to the earlier results of \cite{entmie}, the class of
interactions we consider contains
{\it physically much more realistic potentials}.
Moreover our results for the first time {\it link the Israel-Bishop-Phelps
theorem with classes of interactions whose Fourier transforms have certain
prescribed properties}. Attempts to read off the behaviour of the long-range
correlation functions and the phase transition from the properties of the
Fourier transform of an interaction are of course widespread; they form the
foundation of spin-wave theory. A rigorous version of these arguments holds in
fact in the mean-field limit \cite{pistho}, but for models with interactions
of shorter range it is unclear under which conditions such arguments can be
justified. Our results support the validity of some of such considerations,
if only in a rather implicit fashion.
Similarly to what was observed in \cite{EHM,ent96,zeg94}, we find that there
is a suggestive resemblance between the properties of a non-periodically
ordered system and the presumed properties of spin-glass models.
\bigskip
{\em Acknowledgements}: A.C.D.v.E. thanks Imperial College for its
hospitality. His interest in non-periodic long-range order, as well as some
of the ideas used here, owe much to earlier collaborations and discussions
with Bert Hof, Jacek Mi\c ekisz and Charles Radin.
This research has been supported by EU contract CHRX-CT93-0411.
%\section{Basic Set-up}
\addcontentsline{toc}{section}{\bf References} %\bibliography{potts}
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\end{thebibliography}
\end{document}