\documentstyle[12pt]{article}
\title{How should one define a (weak) crystal?}
\author{A. C. D. van Enter \\ Institute for Theoretical Physics,
P. O. Box 800 \\ University of Groningen\\ Groningen, The
Netherlands \\ and \\ 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}
\pagenumbering{arabic}
\begin{document}
\baselineskip=26pt
\maketitle
{\bf Abstract.} We compare two proposals for the study of
positional long-range order: one in terms of the spectrum of the
translation operator, the other in terms of the Fourier spectrum.
We point out that only the first one allows for the consideration
of molecular, as opposed to atomic, (weakly) periodic structures.
We illustrate this point on the Thue-Morse system.
\\
KEY WORDS: Positional long-range order; Fourier spectrum;\\
spectrum of the shift.
\eject
One of the fundamental questions of statistical mechanics is to
describe ground states of systems of many interacting particles.
It is an important problem to decide if, and to what extent,
ground states are ordered, crystalline or otherwise
\cite{rad1,rad2}. For the description of positional long-range
order, which is necessary for the study of both crystals
(periodic structures) and non-crystalline ordered structures like
quasicrystals \cite{sht,stl}, "turbulent" crystals \cite{rue1},
and "weakly periodic structures" \cite{aub1} there have been two different
main approaches in the literature, both in terms of spectral properties.
In the first one, which has been advocated within the infinite-volume
approach
to statistical mechanics \cite
{rue1,rue2,rue3,rue4,kast,emch1,emch2,bratt}, one
considers the spectrum of the Euclidean group acting as unitary operators
on an
appropriate Hilbert space. For classical models this is the
$L^{2}(\mu)$ space for a translation-invariant Gibbs measure or
ground state measure $\mu$. To be more precise, we will discuss classical
lattice gas models, in which every site of a lattice $Z^{d}$ can
be occupied by one of two (or more generally a finite
number of) different particles (one may also think about them as different
orientations of a spin
variable). An infinite lattice configuration is an assignment of particles
to lattice sites, that is
an element of $\Omega = \{-1,+1\}^{Z^{d}}$. We will be mostly concerned
with
configurations which have uniformly defined frequencies for all finite
patterns. More precisely, to find the frequency of a finite pattern 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
pattern. The closure of the orbit
under translation of any such configuration supports exactly one ergodic
translation-invariant
measure on $\Omega$, say $\mu$, which is uniquely specified by the
frequencies of all finite patterns.
Such systems are called uniquely ergodic. The trivial example is the Ising
antiferromagnet, where apart from the infinitely many defect ground state
configurations there are two
alternating ground state configurations but only one translation-invariant
ground state measure which
is just their average. Let us recall that a sequence of configurations
converges if for any finite
subset of the lattice all but a finite number of configurations are the same
when
restricted to this
subset. Ergodicity of the measure means that for any local observable $f$
the integral $\int f d \mu
\equiv \mu(f)$, i.e., the average of the observable, is equal to $\lim_{n
\rightarrow
\infty}1/N\sum_{i=1}^{N}f(T(x))$, where $T$ is a shift operator, for
$\mu$-almost all $x$. For a
uniquely ergodic measure the limit is uniform with respect to all choices
of $x$. A
uniquely ergodic measure can be a zero temperature limit of a low
temperature Gibbs state (an
infinite-volume grand canonical probability distribution) for an
appropriate interaction. On the
other extreme there is the example of the product (or Bernoulli) measure
obtained by taking
independently every spin to be $+$ or $-$ with probability $1/2$ at every
site. This
translation-invariant measure corresponds to taking infinite temperature.
With respect to this
measure the frequency of plus spins equals $1/2$ but has unbounded
fluctuations, as large regions of
pluses (and in fact any finite configuration) occur with finite (though
small) density. In a typical
"random" sequence these large fluctuations will certainly occur at various
positions hence the
closure of its orbit is the whole configuration space $\Omega$.
If the system is uniquely ergodic (a property which is generic for
ground states \cite{rad3}) the support of its unique ergodic measure
contains one minimal set, i.e., a
non-empty, closed, translation-invariant subset of $\Omega$ which does not
contain any proper
subsets with this property. Physically, this means that all configurations
containing defects have
been eliminated. An example of such a configuration is a ground state
configuration of the
one-dimensional Ising antiferromagnet with spins being $\sigma
_{n}=(-1)^{n}$ for $n \geq 0$ and
$\sigma _{n}= (-1)^{n+1}$ for $n<0$. The defect here is between $-1$ and
$0$. The property of
minimality has been advocated by Aubry \cite{aub1} under the name "weak
periodicity" and has been
discussed in \cite{bomb} under the name "recurrence". Minimality means also
that any pattern in a
sequence appears again within a bounded distance. However, as asserted by
Radin \cite{rad4} in this
connection, this property of minimality or even unique ergodicity does not
imply anything in the
sense of long-range order either in the sense of mixing properties
(asymptotic independence at large
distances) or spectral properties, as the Jewett-Krieger theorem tells us
that any ergodic measure
(including highly disordered ones) is isomorphic to a uniquely ergodic one
\cite{jewe,krieg,denk}.
One possibility of describing
positional long-range order is to consider spectral properties of the shift
operator $T$ (considered
as a unitary operator on $L^{2}(\mu)$). $L^{2}(\mu)$ is the space of
functions which
are square-integrable with respect to $\mu$ and is spanned by sums and
products of spin values at
finite sets of lattice sites. We write $Tf(x)=f(T(x))$. If the support of
$\mu$ is small, many of
these products coincide, viewed as elements of $L^{2}(\mu)$. For example,
in the one-dimensional Ising
antiferromagnet $\sigma_{0}\sigma_{2k}=1$ for both periodic ground state
configurations and
any $k$, hence $\sigma_{0}\sigma_{2k}$ and $1$ are in the same equivalence
class. In fact,
$L^{2}(\mu)$ is two-dimensional here and is spanned by $\sigma_{0}$ and
$1$. The simplest case of a
one-dimensional lattice has been widely studied in ergodic theory
\cite{qef}. We recall that for the
study of the spectrum of the shift operator one considers generalized
2-point functions of the form
$C_{f}(n)=\mu(fT^{n}f)$, where $f$ can be any function in $L^{2}(\mu)$. By
Bochner's theorem one can
write $C_{f}(n)= \int_{0}^{2 \pi}exp(2\pi i\lambda n) m_{f}(d \lambda)$,
where $m_{f}$ are measures
on the interval $[0, 2\pi]$. If we consider higher dimensional lattices we
get measures on $[0,
2\pi]^{d}$ and if we replace a lattice by a continuous space, measures on
$R^{d}$. If for any choice
of $f$, $m_{f}$ has a pure point, singular continuous, or absolutely continuous
part in its Lebesgue
decomposition $m_{f}=m_{f \; pp} + m_{f \; sc} + m_{f \; ac}$, we say that
the spectrum of the shift
has a pure point, singular continuous, or absolutely continuous component.
If the point spectrum
consists solely of finitely many points one interprets this as a crystal;
if one has a dense point
spectrum one has a quasicrystal; and if there is also some (singular)
continuous spectrum one might
have a "turbulent" crystal \cite{rue1}. When there is no long-range order
only absolutely continuous
spectrum is expected. However, there is always one discrete point in the
spectrum at the origin. This
is due to the fact that the constants are translation-invariant and thus
are eigenfunctions of the
shift operator.
In the second approach, which has been pursued by many people
interested in the theory of quasicrystals, see for example
\cite{aub2,aub3,aub4,bomb,tay,savit1,savit2,kolar} one considers
the Fourier spectrum (also called a structure factor or a correlation
measure). This begins with the fact that
diffraction patterns are given by the Fourier transform of the density of
the diffracting matter.
A structure factor $I(\overline{k})$ is proportional to the intensity of a
spot $\overline{k}$ in the
diffraction pattern and for one-dimensional systems $I(k)=
|\sum_{n=-N}^{N}f(n)exp(2\pi i k
n)|^{2}$, where $f(n)$ is an atomic scattering factor of an atom at a
position $n$. For uniquely
ergodic systems it can be shown \cite{qef} that $dkI(k)/(2N+1)$ converges
weakly, as $N \rightarrow
\infty$, to the so called correlation measure which is a spectral measure
$m_{f}$ of the shift
operator corresponding to a special choice of $f=f(0)$ which evaluates
sequences of atomic
scattering factors at the origin (if we have only one type of atoms, $f(n)$
measures the presence of
an atom at a position $n$ and can be written
$f(n)=\frac{1}{2}(\sigma_{n}+1)$). $m_{f}$ is then the
Fourier transform of the 2-point function $\mu(f(0)f(n))$:
$\mu(f(0)f(n))=\int_{0}^{2\pi}exp(2\pi
ikn)m_{f}(dk)$, where $\mu$ is the unique ergodic measure supported by the
closure of the orbit of a
configuration of diffracting matter. $m_{f}$ is a measure on the interval
$[0,2\pi ]$ and it will
have a (quasi)crystalline character if it consists solely of (in)finitely
many points in the pure
point part of its Lebesgue decomposition.
We first point out that these two notions are not equivalent. We
will illustrate this point on the example of the Thue-Morse
system. Moreover, we observe that the first characterization has
the advantage of allowing us to recognize molecular structures.
To define the Thue-Morse system we start by taking a sequence of
all $+$ spins ($\sigma_{i} = \pm 1, i \in Z$). At the first step we
flip every second spin. At the $n th$ step we flip all blocks of
$2^{n-1}$ spins within the previous $n-1 st$ configuration from the
site $(2k+1)2^{n-1}+1$ to $(2k+2)2^{n-1}$ for every $k$.
A cluster
point of this sequence of periodic configurations of period
$2^{n}$ is a nonperiodic sequence called a Thue-Morse
sequence. Its first 32 elements are
$$+ - - + - + + - - + + - + - - + - + + - + - - + + - - + - + + - .$$
The closure of its orbit under translation supports
exactly one ergodic translation-invariant measure $\mu_{TM}$ \cite{kea}.
The Thue-Morse measure $\mu_{TM}$ was shown to be a unique
ground state for arbitrarily fast decaying 4-body interactions
\cite{fra1}. Some aspects of its non-conventional long-range
order were shown to persist at finite temperatures for some
slowly decaying but still summable 4-body interactions
\cite{fra2}. In fact, a Thue-Morse sequence has been
experimentally realized \cite{merl,axel} and the spectrum was
observed to be singular continuous.
If one considers the Fourier transform of
$C(n)=\mu_{TM}(\sigma_{0}\sigma_{n})$ it is known to be singular
continuous \cite{mahl,kaku,kea,qef}. Thus looking at the Fourier
spectrum there would not be any quasicrystalline structure.
However, if one considers the shift operator acting on the whole
space $L^{2}(\mu_{TM})$ we find a richer behavior \cite{kea}. In
particular, we have: $L^{2}(\mu_{TM})=L^{2,odd}(\mu_{TM}) \oplus
L^{2,even}(\mu_{TM})$, where the $L^{2,odd/even}(\mu_{TM})$
spaces are spanned by odd/even functions with respect to the spin
flip operator $\sigma_{i} \rightarrow -\sigma_{i}$. $T$ acting on
$L^{2,odd}(\mu_{TM})$ has a singular continuous spectrum, while
acting on $L^{2,even}(\mu_{TM})$ it has a dense point spectrum.
Thus, if we look at correlation functions
$C_{f}(n)=\mu_{TM}(fT^{n}f)$ with $f$ even, for example
$f=\sigma_{0}\sigma_{1}$, or more generally $f=\sigma^{X}$, $|X|$
even, its Fourier transform consists only of points ($\delta-
peaks)$. The special choice $f=\sigma_{0}$ gives a singular
continuous spectrum because of the oddness of $\sigma_{0}$. If we
consider an $X$ to be the shape of a molecule, these molecules
can have $\delta -peaks$ and can form more ordered structures than the
underlying atoms. To be more precise: if $\omega$ is any
Thue-Morse sequence then $s^{n}_{A}=T^{n}\sigma^{A}(\omega)$ for
$|A|$ even is a q-periodic sequence according to the following
definition \cite{rad5}: A configuration of particles is
q-periodic if, when a certain fraction of them is ignored, the
rest of the configuration is periodic; the smaller the fraction
the larger the period. By construction a Thue-Morse sequence
is a sequence of blocks of spins $M_{k}$ of length $2^{m}$ such that
$M_{k}= \pm M(m)$ for some fixed block configuration $M(m)$ and this
holds for all $m\geq 0$. Now, if one ignores lattice sites at the
boundaries between consecutive $M_{k}'s$ (the density of which is
$(diam(A)-1)/2^{m})$ then $s^{n}_{A}$ is a part of a periodic
configuration of period $2^{m}$, which shows that $s^{n}_{A}$ is
q-periodic.
As an example, if one places a
delta function weight midway between each pair of equal adjacent spins and
takes the Fourier
transform of this object, it will consists entirely of delta peaks,
whereas, if a delta function is
placed midway between each pair of $+$ neighboring spins, the Fourier
transform will have both delta
peaks and a continuous part, and finally, if delta functions are placed on
the $+$ spins and omitted
from the $-$ spins, the Fourier transform will have no delta peaks except
the one at the origin.
If a "molecule" is a pair of two neighboring equal spins the molecular
spectrum is
quasiperiodic, as $\delta_{\sigma_{0}\sigma_{1}}$ is even, while a
"molecule" consisting of two +
neighbors gives rise to a mixed spectrum because
$f=1/4(\sigma_{0}+1)(\sigma_{1}+1)=1/4(\sigma_{0}\sigma_{1}+1+
\sigma_{0}+\sigma_{1})$ has both an
even and an odd part. Finally, placing delta functions just on the $+$
spins (the atoms) corresponds
to taking $f=\frac{1}{2}(\sigma_{0}+1)$ which is a sum of an odd function
and a constant producing
only a $\delta-peak$ at the origin.
For an example of a 2-dimensional system which has no $\delta -
peaks$, at either atomic or molecular level, see
\cite{rad4} following \cite{moz}. This is in fact a unique ground state
for a classical lattice gas model with a nearest neighbor interaction.
As for positive temperatures, there exists a 3-dimensional finite
range ferromagnetic model due to Slawny \cite {slaw} of a mixing
Gibbs state, thus $\mu(fT^{n}f) \rightarrow (\mu(f))^{2}$ for all
$f$, which has long-range order in the sense that it is not an
extremal Gibbs state. It is not 3-mixing, that it is to say
$\mu(fT^{n_{1}}fT^{n_{2}}f) \not\rightarrow (\mu(f))^{3}$ for
some $f$ and $|n_{1}|, |n_{2}|$ growing such that also $|n_{1}-n_{2}|
\rightarrow \infty$. This is an example of Ruelle's
\cite{rue1} proposed definition of a "turbulent" crystal, a
system in which the decomposition into extremal Gibbs states is
strictly finer than the almost periodic decomposition connected
with the discrete part of the spectrum of the shift operator.\\
\vspace{5mm}
{\bf Acknowledgments.} We thank R. B. Griffiths, M. Winnink, and
A. Hof for discussions. Our interest in these issues owes much
to previous collaborations with C. Radin. We thank Universit\'{e}
Catholique de Louvain-la-Neuve and the Rijksuniversiteit
Groningen for their hospitality during visits. The research of A.
C. D. van Enter has been made possible by a fellowship of the
Royal Netherlands Academy of Arts and Sciences and of J.
Mi\c{e}kisz by Bourse de recherche UCL/FDS.
\begin{thebibliography}{99}
\bibitem{rad1} C. Radin, {\em Low temperature and the origin of
crystalline symmetry,} {\em Int. J. Mod. Phys.} {\bf B1}: 1157
(1987).
\bibitem{rad2} C. Radin, {\em Global order from local sources,} to appear
in
October, 1991 issue of {\em Bull. Amer. Math. Soc.}
\bibitem{sht} D. Shechtman, I. Blech, D. Gratias, and J. W.
Cahn, {\em Metallic phase with long-range orientational order and
no translation symmetry,} {\em Phys. Rev. Lett.} {\bf 53}: 1951
(1984).
\bibitem{stl} D. Levine and P. J. Steinhardt, {\em Quasicrystals:
A new class of ordered structures,} {\em Phys. Rev. Lett.} {\bf
53}: 2477 (1984).
\bibitem{rue1} D. Ruelle, {\em Do turbulent crystals exist?} {\em
Physica} {\bf 113A}: 619 (1982).
\bibitem{aub1} S. Aubry, {\em Devil's Staircase and order without
periodicity in classical condensed matter,} {\em J. Physique}
{\bf 44}: 147 (1983).
\bibitem{rue2} D. Ruelle, {\em Statistical Mechanics; Rigorous
Results}, (especially Ch. 6) (Benjamin, Reading Ma, 1969).
\bibitem{rue3} D. Ruelle, {\em States of physical systems,} {\em
Commun. Math. Phys.} {\bf 3}: 133 (1966).
\bibitem{rue4} D. Ruelle, {\em Integral representation of states
on a $C^{*}$ algebra,} {\em J. Funct. Anal.} {\bf 6}: 116 (1970).
\bibitem{kast} D. Kastler and D. W. Robinson, {\em Invariant
states in statistical mechanics,} {\em Commun. Math. Phys.} {\bf
3}: 151 (1966).
\bibitem{emch1} G. G. Emch, {\em The $C^{*}$-algebraic approach
to phase transitions,}in {\em Phase Transitions and Critical
Phenomena} Vol. 1 Eds. C. Domb and M. L. Green, (Academic Press,
London-New York, 1972).
\bibitem{emch2} G. G. Emch, H. J. F. Knops, and E. J. Verboven,
{\em Breaking of Euclidean symmetry with an application to the
theory of crystallization,} {\em J. Math. Phys.} {\bf 11}: 1165
(1970).
\bibitem{bratt} O. Bratteli and D. W. Robinson, {\em Operator
Algebras and Quantum Statistical Mechanics} Vol 1 and 2
(Springer, Berlin Heidelberg New York, 1979,1981).
\bibitem{rad3} C. Radin, {\em Correlations in classical ground
states,} {\em J. Stat. Phys.} {\bf 43}: 707 (1986).
\bibitem{bomb} E. Bombieri and J. E. Taylor, {\em Quasicrystals,
tilings, and algebraic number theory,} in {\em Contemporary
Mathematics} Vol. 64 (1987).
\bibitem{rad4} C. Radin, {\em Disordered ground states
of classical lattice models,} {\em Rev. Math. Phys.} {\bf 3}: 125 (1991).
\bibitem{jewe} R. I. Jewett, {\em The prevalence of uniquely
ergodic systems,} {\em J. Math. Mech.} {\bf 19}: 717 (1970).
\bibitem{krieg} W. Krieger, {\em On unique ergodicity,} in
the Proceedings of the Sixth Berkeley Symposium on Mathematical
Statistics and Probability, pp 327-346 (1970).
\bibitem{denk} M. Denker, C. Grillenberger, and K. Sigmund, {\em
Ergodic Theory on Compact Spaces,} (Springer LNM {\bf 527},
Springer, Berlin Heidelberg New York, 1976).
\bibitem{qef} M. Queff\'{e}lec, {\em Substitution Dynamical Systems -
Spectral Analysis} (Springer LNM {\bf 1294}, Springer, Berlin
Heidelberg New York, 1987).
\bibitem{aub2} S. Aubry, {\em Weakly periodic structures and
example,} {\em J. Physique Colloque} {\bf C3} Tome 50: 97 (1984).
\bibitem{aub3} S. Aubry, {\em Weakly periodic structures with a
singular continuous spectrum,} Proceedings of the NATO Advanced
Research Workshop on Common Problems of Quasi-Crystals, Liquid-
Crystals and Incommensurate Insulators, Preveza 1989. Ed. J. I.
Toledano.
\bibitem{aub4} S. Aubry, C. Godr\'{e}che, and J. M. Luck, {\em
Scaling properties of a structure intermediate between
quasiperiodic and random,} {\em J. Stat. Phys.} {\bf 51}: 1033
(1988).
\bibitem{tay} J. W. Cahn and J. E. Taylor, {\em An introduction
to quasicrystals,} in {\em Contemporary Mathematics} Vol. 64 (1987).
\bibitem{savit1} Z. Cheng, R. Savit, and R. Merlin, {\em Structure
and electronic properties of Thue-Morse lattices,} {\em Phys.
Rev.} {\bf B37}: 4375 (1988).
\bibitem{savit2} Z. Cheng and R. Savit, {\em Structure factor of
substitutional sequences,} {\em J. Stat. Phys.} {\bf 60}: 383
(1990).
\bibitem{kolar} M. Kolar, M. K. Ali, and F. Nori, {\em
Generalized Thue-Morse chains and their physical properties,}
{\em Phys. Rev.} {\bf B43}: 1034 (1991).
\bibitem{kea} M. Keane, {\em Generalized Morse sequences,} {\em Zeit.
Wahr.}
{\bf 10}: 335 (1968).
\bibitem{fra1} C. Gardner, J. Mi\c{e}kisz, C. Radin, and A. C. D.
van Enter, {\em Fractal symmetry in an Ising model,} {\em J.
Phys.} {\bf A22}: L1019 (1989).
\bibitem{fra2} A. C. D. van Enter and J. Mi\c{e}kisz, {\em
Breaking of periodicity at positive temperatures,} {\em Commun.
Math. Phys.} {\bf 134}: 647 (1990).
\bibitem{merl} R. Merlin, K. Bajemu, J. Nagle, and K. Ploog, {\em
Raman scattering by acoustic phonons and structural properties
of Fibonacci, Thue-Morse, and random superlattices,} {\em J.
Physique Colloque} {\bf C5}: 503 (1987).
\bibitem{axel} F. Axel and H. Terauchi, {\em High resolution X-
ray diffraction spectra of Thue-Morse GaAs-AlAs heterostructures:
towards a novel description of disorder,} {\em Phys.
Rev. Lett.} {\bf 66}: 2223 (1991).
\bibitem{rad5} C. Radin, {\em Crystals and quasicrystals: A
lattice gas model,} {\em Phys. Lett.} {\bf 114A}:385 (1986).
\bibitem{mahl} K. Mahler, {\em On the translation properties of a
simple class of arithmetical functions,} {\em J. Math and Phys.}
{\bf 6}: 150 (1927).
\bibitem{kaku} S. Kakutani, {\em Ergodic properties of shift
transformations,} in the Proceedings of the Fifth Berkeley Symposium
on Mathematical Statistics and Probability, pp 404-414 (1967).
\bibitem{moz} S. Mozes, {\em Tilings, substitutions, and
dynamical systems generated by them,} {\em J. d'Analyse Math.}
{\bf 53}: 139 (1989).
\bibitem{slaw} J. Slawny, {\em Ergodic properties of equilibrium
states,} {\em Commun. Math. Phys.} {\bf 80}: 477 (1981).
\end{thebibliography}
\end{document}