\documentstyle[12pt]{article}
\title{Overlap Distributions for Deterministic Systems with Many Pure
States}
\author{A. C. D. van Enter \\ Institute for Theoretical Physics, P. O. Box
800
\\ University of Groningen\\ Groningen, The Netherlands \\ \\ A.~Hof \\
Departement
Mathematik \\ ETH-Zentrum \\ 8092 Z\"{u}rich, Switzerland \\ \\ Jacek
Mi\c{e}kisz \\ Institut de
Physique Th\'{e}orique \\ Universit\'{e} Catholique de Louvain \\ B-1348
Louvain-la-Neuve, Belgium}
\pagenumbering{arabic}
\begin{document}
\baselineskip=26pt
\maketitle
{\bf Abstract.} We discuss the Parisi overlap distribution function for
various deterministic
systems with uncountably many pure ground states. We show examples of
trivial,
countably discrete, and continuous distributions. \eject
In Parisi's proposed solution for the Sherrington-Kirkpatrick spin-glass
model \cite{par1,par2,par3}
there occurs an overlap distribution $p(q)$ which is non-trivial in the
sense that it has a
continuous part and two delta functions. The suggestion is that such a
non-trivial $p(q)$ represents
the presence of infinitely many pure extremal Gibbs or ground states.
As the mathematical status of Parisi's theory is yet ill-understood, it
seems of interest to study
its various aspects in simpler models. For instance, Fisher and Huse
\cite{fh1} have studied the
behaviour of the overlap distribution and discussed its strong dependence
on boundary conditions in
various examples with only two pure states. Their work was partly motivated
by their conjecture that
short-ranged spin-glasses have only two pure states \cite{fh2,fh3,fh4}.
We present here some results from a complementary point of view and
consider what might happen in
deterministic systems with infinitely many pure states. We suspect that
some spin glass models
do indeed have infinitely many ground states \cite{bf,ae1}, although we
consider this matter
unsettled at present. The scenario with many states was recently considered
by Newman and Stein
\cite{ns}. In fact, our paper originated from discussions with C. Newman.
Despite somewhat different
physical motivations, our conclusions support those of Fisher and Huse: the
overlap distribution does
not describe the number of states well.
The reason for this conclusion differs between our examples and theirs.
Whereas Fisher and Huse show
that standard boundary conditions (free, periodic, antiperiodic) can either
suppress some pure state
(as happens for example in the random field Ising model) or give rise to a
continuous overlap
distribution due to floating defects (as happens for example in the nearest
neighbour ferromagnet
with antiperiodic boundary conditions), in our examples we work with states
which are mixtures of
uncountably many pure states that are free of defects. Hence no pure state
is suppressed, and floating
defects do not occur. As we work directly with the infinite-volume
measures, we need not consider
boundary conditions. Our systems are deterministic in the sense that their
configurations are
generated by deterministic rules: standard substitution rules producing
Thue-Morse and Fibonacci
sequences. Also, our configurations are ground state configurations of
deterministic
translation-invariant interactions.
More specifically, in our models at every site $i$ of the one-dimensional
lattice $\bf Z$ there is a
spin variable $\sigma_{i}$ which can attain the values $\pm 1$. An infinite
lattice configuration is
an assignment of spin orientations to lattice sites, that is an element of
$\Omega=\{-1,+1\}^{Z}$. We
are concerned with nonperiodic configurations which have nevertheless
uniformly defined frequencies
for all finite patterns.
These are examples of the so-called similar but incongruent pure phases
discuss ed in
\cite{fh1}.
More precisely, to find a frequency of a finite pattern in a given
configuration we first count the number of times it appears in a segment of
size $l$ and centered at
the origin of the lattice, divide it by $l$, and then take the limit $l
\rightarrow \infty$. If
the convergence is uniform with respect to the position of the segments
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 strictly ergodic if every finite pattern that occurs in
the configuration occurs
with a uniformly defined frequency that is strictly larger than zero. The
measures we consider are
strictly ergodic. Strictly ergodic measures can be considered to be the
typical ground states for
translation-invariant interactions \cite{aub1,rad1,aub2,rad2}.
Let us denote by $q_{XY}$ the overlap between two configurations $X$ and
$Y$ in the support of $\mu$. It is defined by
$$q_{XY}=lim_{N\rightarrow\infty}(1/N)\sum_{i=1}^{N}\sigma_{i}(X)\sigma_{i}(
).$$
Then the Parisi
overlap distribution $p(q)$ is the distribution of $q_{XY}$ with respect to
the product measure
$\mu\otimes \mu$.
Our first result is a simple application of a well known result from
ergodic theory. It concerns
the so called weakly-mixing measures. Let us recall that a measure $\mu$ is
weakly mixing if
$\mu(fT^{k(n)}f) \rightarrow [\mu(f)]^{2}$ for all $f$ square-integrable
with respect to $\mu$, where
$T$ is a shift operator, and $k(n)$ the sequence of natural numbers,
possibly excluding a set of
zero density (depending on $f$). This property is equivalent to $T$ having
a
continuous spectrum \cite{wal}.
\newtheorem{dodo}{Theorem}
\begin{dodo}
If $\mu$ is weakly mixing then $p(q)$ is a point distribution concentrated
on
$[\mu(\sigma_{0})]^{2}.$
\end{dodo}
{\bf Proof:} If $\mu$ is weakly mixing then $\mu \otimes \mu$ is ergodic
\cite{wal} and by the
ergodic theorem $(1/N)\sum_{i=1}^{N}\sigma_{i}(X)\sigma_{i}(Y)$ converges
with probability one to
$[\mu(\sigma_{0}]^{2}.$ $\Box$\medskip
A specific weakly mixing example of a 3-dimensional ferromagnetic Ising
model wi th
uncountably many Gibbs states and
a trivial overlap distribution was already given in \cite{ja}. However, in
that example all pure
states are related by a global symmetry of the system. This is not the case
in the models considered
here.
Our next result answers a question of C. Newman about the Thue-Morse
system. To define
the Thue-Morse system we start by taking a sequence of all $+1$ spins. 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 $2\cdot 2^{n}$ is a
nonperiodic sequence called
a Thue-Morse sequence \cite{kea,aub3,luck,my1,my2,my3}. The closure of its
orbit under translation
supports exactly one translation-invariant measure $\mu_{TM}$ which is in
fact strictly ergodic
\cite{kea}. Thue-Morse sequences can also be obtained by iterating the
following substitution rule:
$1 \rightarrow 1 -1$, $-1 \rightarrow -1 \; 1$. The Thue-Morse measure
$\mu_{TM}$ has been shown to
be the unique ground sta te for arbitrary rapidly decaying 4-spin
interactions \cite{my1}.
\begin{dodo}
The overlap distribution $p(q)$ for $\mu_{TM}$ is a point measure
concentrated on $q=0$.
\end{dodo}
{\bf Proof:} $L^{2}(\mu_{TM})$, the space of functions which are
square-integrable with respect to
$\mu_{TM}$ can be decomposed into the direct sum of the two spaces spanned
by the odd and the even
functions
with respect to the spin-flip operator $\sigma_{i} \rightarrow
-\sigma_{i}.$ The shift operator
acting on the space of odd functions has a singular continuous spectrum
\cite{kea}. Therefore
when
you consider only odd observables, like $\sigma_{i}$, then $\mu_{TM}$
behaves as if it were weakly
mixing and
$\mu_{TM} \otimes \mu_{TM}$ as if it were ergodic with respect to these
observables
and so the conclusion follows
as in the proof of the Theorem 1. $\Box$\medskip
Now, let $X$ be any Thue-Morse sequence and let $Y(i)=X(i)\cdot X(i+1).$
The closure of the orbit
of $Y$ obviously supports exactly one ergodic translation-invariant measure
and the resulting
strictly ergodic system is called a Toeplitz system \cite{dek,jk}. Every
Toeplitz sequence can be
constructed in the following way. First choose a sublattice $L_1$ of
period~2 and put a -1 on every
site in $L_1$. Next, choose a sublattice $L_2$ of period~4 that is disjunct
from $L_1$ and put a +1
on every site in $L_2$. In this way one continues: $L_j$ is a sublattice of
period~$2^j$ that is
disjunct from $L_1,\ldots,L_{j-1}$ and the spins in $L_j$ are $(-1)^j$. In
the interpretation of
\cite{my3} the Toeplitz sequence describes the molecules of the Thue-Morse
system.
\begin{dodo}
The overlap distribution for the Toeplitz system $\mu_{T}$ contains
countably many points.
\end{dodo}
{\bf Proof:} We will fix one Toeplitz configuration $Y$ and calculate its
overlaps with all Toeplitz
sequences grouped with respect to the constant overlap. First consider all
Toeplitz configurations
such that the minuses of the first sublattice are exactly off the first
sublattice of $Y$. This gives
rise to a point measure of mass $1/2$ concentrated on $q=-1/2+1/4-1/8+...=-
1/3$. Now consider all
Toeplitz configurations such that minuses of the first sublattice are on
the first sublattice of $Y$
and the pluses of the second sublattice are off the second sublattice of
$Y$. This gives us a point
measure of mass $1/4$ concentrated on $q=1/2-1/2\cdot 1/3=1/3.$ In the next
step the first two
sublattices are coinciding and the third ones miss each other, giving rise
to a point measure with
mass $1/8$ and concentrated on $q=1/2+1/4-1/4\cdot 1/3=2/3.$ Repeating this
procedure infinitely many
times we obtain $p(q)=\sum_{n=0}^{\infty}(1/2^{n+1})\delta\Big(q-(3\cdot
2^{n-2} -
1)/(3\cdot 2^{n-2})\Big).$
$\Box$\medskip
Note that this construction resembles the ``ultrametric'' structure that
occurs in Parisi's theory
\cite{par2}.
Our last example is the Fibonacci system. It too is strictly ergodic. We
will show that it has a
continuous part in its overlap distribution. A Fibonacci sequence can be
obtained using the following
substitution rules: $1 \rightarrow - 1$, $-1 \rightarrow -1 \; 1$.
\begin{dodo}
The overlap distribution of the Fibonacci system $\mu_{F}$ has a continuous
part.
\end{dodo}
{\bf Proof:} We will use an equivalent construction of Fibonacci sequences
by rotations $T$ over
the circle by an amount $2\pi \gamma$ with $\gamma=2/(1+\sqrt5)$ being the
golden ratio (see e.g.\
\cite{sch}). To every angle $2\pi \phi \in [0,2\pi )$ there corresponds a
Fibonacci sequence $X$ in
the following way. If $T^{n}\phi$ is in the arc segment $[0,2\pi\gamma)$
then $X(n)=1$, otherwise
$X(n)=-1.$ Now, because of the irrationality of $\gamma$, $T$ is ergodic
with respect to the Lebesgue
measure $\mu_{L}$ on the circle. Hence the overlap between sequences
corresponding to $2\pi \phi_{1}$
and $2\pi \phi_{2}$ depends only on $\alpha=\phi_{1}-\phi_{2}$. Namely,
$q(\alpha)=\mu_{L}(A)-\mu_{L}(A^{C})$, where $A$ is the event where the two
line segments that define
the angle $2\pi \alpha$ are both in the same arc segment $[0,2\pi \gamma)$
or $[2\pi \gamma, 2\pi)$.
Hence \[ q(\alpha) = \left\{ \begin{array}{lll} 1-4\alpha & \mbox{if $0
\leq \alpha < 1-\gamma$} \\
1-4(1-\gamma) & \mbox{if $1-\gamma \leq \alpha < \gamma$}\\ 1-4(1-\alpha) &
\mbox{if $\gamma \leq
\alpha < 1$ }
\end{array}
\right. \]
It follows that
$p(q)= (2\gamma-1) \delta\Big(q-(1-4(1-\gamma)\Big)
+ \frac12 1_{[1-4(1-\gamma),1]}(q)dq$. $\Box$\medskip
It is not known if there is a simple (e.g.\ finite-spin exponentially
decaying) translation-invariant
interaction with $\mu_{F}$ as its unique ground state, although by
\cite{aub2,rad2} there are
infinite-spin interactions for which $\mu_{F}$ is the unique ground state.
Let us remark here that
such a deterministic interaction has a continuous part in its overlap
distribution, a property usually
attributed to systems with random interactions like spin glasses.
Let us mention that overlaps have been studied in the literature on
substitution dynamical systems
\cite{dek,mich,quef} under the name ``coincidence density''. However, not
much seems to be known
about their distributions. The overlap between two finite sequences of
$\pm1$ is a linear function of
their Hamming distance.
We also remark that the Edwards-Anderson parameter as for example studied
in \cite{aeg}
measures the maximal overlap, and hence would be $1$ in our examples. This
shows that there can be a
big difference between a maximal and a typical overlap.
Concluding, we have shown that in various examples where one can compute
the overlap distribution
for systems with infinitely many states, various types of distributions
occur. Thus overlap
distributions do not provide a good description of the number of pure
phases of the system. The
fact that we worked at
$T=0$ should not matter too much as similar nonperiodic long range order
and infinitely many pure
Gibbs states can occur at positive temperatures \cite{my2,izra}. This
conclusion fully supports what
Fisher and Huse found in their examples with finitely many states and
suggests for spin-glass models
that the overlap distribution also there might not be a very useful
quantity.
{\bf Acknowledgments.} We thank Michel Dekking, Chuck Newman, and Marinus
Winnink for discussions.
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{par1} Parisi G 1983 {\em Phys. Rev. Lett.} {\bf 50} 1946
\bibitem{par2} Mezard M, Parisi G,
and Virasoro M A 1987 {\em Spin Glass Theory and Beyond} (Singapore: World
Scientific)
\bibitem{par3}
Sherrington D and Kirkpatrick S 1975 {\em Phys. Rev. Lett.} {\bf 35} 1792
\bibitem{fh1} Huse D A and
Fisher D S 1987 {\em J. Phys.} {\bf A20} L997
\bibitem{fh2} Fisher D S and Huse D A 1987 {\em J. Phys.} {\bf A20} L1005
\bibitem{fh3} Fisher D S and Huse D A 1986 {\em Phys. Rev. Lett.} {\bf 56}
1601
\bibitem{fh4} Fisher D S and Huse D A 1988 {\em Phys. Rev.} {\bf B30} 386
\bibitem{bf} Bovier A and Fr\"{o}hlich J 1986 {\em J. Stat. Phys.} {\bf 44}
347
\bibitem{ae1} Enter A C D van 1990 {\em J. Stat. Phys.} {\bf 60} 275
\bibitem{ns} Newman C M and Stein D L 1992 {\em Multiple States and
Thermodynamic Limits in
Short-Ranged Ising Spin Glass Models} To appear in Phys.\ Rev. B
\bibitem{aub1} Aubry S 1983 {\em J. Phys. (Paris)} {\bf 44} 147
\bibitem{rad1} Radin C 1986 {\em J. Stat. Phys.} {\bf 43} 707
\bibitem{aub2} Aubry S
1989 {\em J. Phys. (Paris) Coll.} {\bf 50} 97
\bibitem{rad2} Radin C 1991 {\em Rev. Math. Phys.} {\bf3} 125
\bibitem{wal} Walters P 1982 {\em
Introduction to Ergodic Theory} (New York: Springer)
\bibitem{ja} Mi\c{e}kisz J 1989 {\em J. Stat. Phys.} {\bf 55} 351
\bibitem{kea} Keane M 1968 {\em Z.
Wahrs. verw. Geb.} {\bf 10} 335
\bibitem{aub3} Aubry S
1989 {\em Weakly Periodic Structures with a Singular Continuous Spectrum}
in Toledano J I (ed):
{\em Proceedings of the NATO advanced research workshop on Common Problems
in Quasicrystals, Liquid
Crystals, and Incommensurate Insulators}, Preveza, Greece, 4th-8th
September 1989
\bibitem{luck} Luck J M 1989 {\em Phys. Rev.} {\bf B39} 5834
\bibitem{my1} Gardner C, Mi\c{e}kisz J,
Radin C, and Enter A C D van 1989 {\em J. Phys.} {\bf A22} L1019
\bibitem{my2} Enter A C D van and Mi\c{e}kisz J 1990 {\em Commun. Math.
Phys.} {\bf 134} 647
\bibitem{my3} Enter A C D van and Mi\c{e}kisz J 1992 {\em J. Stat. Phys}
{\bf 66} 1147
\bibitem{dek} Dekking F M 1980 {\em
Substitutions} Thesis Nijmegen and private communication
\bibitem{jk} Jacobs K and Keane M 1969 {\em Z. Wahrs. verw. Geb.} {\bf 13} 123
\bibitem{sch} Schr\"{o}der M 1991 {\em Fractals, Chaos, Power Laws} (New
York:
Freeman) p. 308
\bibitem{mich} Michel P 1987 {\em Z. Wahrs. verw. Geb.} {\bf 42} 205
\bibitem{quef} Queff\'{e}lec M 1987 {\em Substitution Dynamical Systems}
(Springer LNM 1294)
\bibitem{aeg} Enter A C D van and Griffiths R B 1983 {\em Commun. Math.
Phys.} {\bf 91} 319
\bibitem{izra} Israel R. B. Private communication
\end{thebibliography}
\end{document}