NOTE: This paper will appear in Reviews of Mathematical Physics
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\hbox{}
\vskip 1truein\centerline{{\bf DISORDERED GROUND STATES OF CLASSICAL LATTICE
MODELS}\footnote*{Research supported in part by NSF Grant No. DMS-9001475\hfil}}
\vskip .5truein\centerline{by}
\centerline{Charles Radin}
\vskip .2truein\centerline{Mathematics Department}
\centerline{University of Texas}
\centerline{Austin, TX\ \ 78712}
\centerline{USA}
\vskip .5truein\centerline{ABSTRACT}
{\narrower\vskip .2truein\noindent We use strictly ergodic dynamical
systems to describe two methods for constructing short range
interactions of classical statistical mechanics models with unique
ground states and unusual properties of disorder; in particular,
these ground states can be mixing under translations (and therefore
have purely continuous spectrum), and can have positive entropy.
Because of the uniqueness of the ground state the disorder is not of
the usual type associated with local degeneracy.\smallskip}
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\nd {\bf 1.\ Introduction.} The goal of this paper is twofold: to
demonstrate that the mathematics of strictly ergodic dynamical
systems can be useful in mathematical physics, and to use it
to get new results on the ``crystal problem''.
The crystal problem is the attempt to understand why real materials
seem to have a strong tendency to be highly ordered at low temperature
and high pressure. The microscopic structure of solids can be quite
complicated. The range of experimentally known structures is much
reduced however by restricting attention to materials in thermal
equilibrium, and rejecting glasses as nonequilibrium, which we do.
Even so one is left, at least, with crystals (with perhaps large unit
cells), incommensurate solids, and quasicrystals [34]. The basic
problem we wish to address is to understand where these structures
come from on the basis of (classical, equilibrium) statistical
mechanics.
This is a notoriously hard problem; see [27] for an historical review.
Much of the progress on this ``crystal problem'' concerns lattice
(toy) models, and in particular the ground states (that is, the zero
temperature states), of lattice models. (There has also been important
work on continuum models, in particular by Aubry and coworkers [2].)
The results have been of two kinds: there have been interesting
examples -- such as those based on nonperiodic tilings of space
[23,25,26,27] -- which exhibit the qualitative properties of
quasicrystals, and there have been some generic results, concerning
the existence of long range order among general classes of models
[15,24,28].
In this paper we will use strictly ergodic dynamical systems
to describe two methods for producing lattice models for which the
ground state has previously unattainable properties of disorder. Various
conventional meanings of the term ``disorder'' will be considered
below.
\vskip .2truein
\nd {\bf 2.\ Notation.} It is
necessary to introduce a framework which reveals the manner in which
interactions influence the structure of ground states. Within
mathematics our framework is called symbolic dynamical systems and
within physics it is known as classical lattice gas models. We need to
review some traditional terminology and results from both
perspectives.
First we summarize some conventional notation from the mathematics of
symbolic dynamical systems. In (topological) dynamical systems one
starts with a compact metrizable space $X$, a Borel probability
measure $m$ on $X$ and a group of homeomorphisms of $X$ which leave
$m$ invariant. The support of $m$ is defined as the complement in $X$
of the union of all open sets of zero measure. A closed subset $Y$ of
$X$, invariant under the homeomorphisms, is called minimal if it contains
no proper closed invariant subset, it is called uniquely ergodic if
there is only one invariant Borel probability measure with support in
$Y$, and it is called strictly ergodic if it is both uniquely ergodic
and minimal. A point $x \in X$ will be called minimal, uniquely
ergodic or strictly ergodic if the closure of its orbit under the
homeomorphisms, $\bar O(x)$, has the property. A dynamical system is
called a d-dimensional symbolic dynamical system if the space $X$ is a
subset of a product space of the form $A^{{\bf Z}^d}$ with the integer
$d\ge 1$, the group of homeomorphisms is that implementing the
translations $\{T^j\ \vert\ j\in{\bf Z}^d\}$, and the set $A$ is
finite. By the ``spectrum'' of (the translations in) a symbolic
dynamical system we refer to the spectrum of the unitary operators
which implement the translations of ${{\bf Z}^d}$ on the complex
Hilbert space $L_2(X,m)$, in the following standard sense. There is
a projection valued measure $dE(\lambda)$ on $[0,1)^d$ such that
for any vectors $f,g \in L_2(X,m),$ $$ \ =
\int_{[0,1)^d}\exp(j\!\cdot\!\lambda\, 2\pi i)\ d\!,
\ \ \ j \in {\bf Z}^d \eqno 1)$$
\noindent where $<\,,>$ refers to the inner product in $L_2(X,m).$
For discussion of the above and other concepts from dynamical systems,
such as the various types of mixing and entropy, see [21,5,20,29].
Next we define the type of statistical mechanical model known as a
classical lattice gas (or spin) model. As physical space we take the
integer lattice ${\bf Z}^d,$ with dimension $d\geq1.$ At each site $j$
in ${\bf Z}^d$ we associate a copy of the finite set $A$ (representing
the possible local states at each site) of cardinality $N(A)\geq 2$.
(One of the elements of $A$ could denote an ``unoccupied'' site.) If we
wish to include an extended hard core condition, the configuration
space, $\Omega$, of the model is obtained from the Cartesian product
$A^{{\bf Z}^d}$ by removing all points $\omega$ (henceforth called
configurations) in which two sites, $j, k$, occupied by particles,
have (Euclidean) separation $\Vert j-k\Vert\leq R$, where the hard
core radius $R\geq 1$ is fixed. (See [33,32] for discussions of hard
core implementation.)
We now need several definitions in order to describe the allowed
interactions of our models. Let $B$ be the set of all nonempty finite
subsets of ${\bf Z}^d.$ We introduce an equivalence relation on $B$ by
declaring two sets $b_1$ and $b_2$ equivalent if there is some
(lattice) translation $T$ such that $T(b_1)=b_2.$ From each
equivalence class we select a representative which, as a set, contains
the origin {\bf 0} of the lattice, and we denote the set of these
representatives by $B_0$. For each $b$ in $B$ we denote by
$\Omega_b$ the set of all possible restrictions $\omega^b$ to $b$
of configurations $\omega$ in $\Omega$, and by $\Omega^r$ the set of
all these restrictions for all $b$ in $B_0$: $\Omega^r \equiv \bigcup_{b\in
B_0} \Omega_b$. (For a singleton $b = \{j\}$ we use the abbreviation
$\omega^j$ in place of $\omega^{\{j\}}$.)
Using the above notation we now introduce interactions
as follows [31,10]. By the (translation invariant) interaction $V$ we
mean a real valued function on $\Omega^r$ -- that is, an assignment of
the (many-body) energy $V(\omega^b)$ to each state of occupancy of
each finite set of sites -- with the restriction that
$$\sum_{\omega^b\in\Omega^r}\vert V(\omega^b)\vert
\ g(\omega^b) \equiv \Vert V\Vert_g < \infty\eqno 2)$$
\nd where $g(\omega^b)$ depends only on the set $b$ on which
$\omega^b$ is defined, and can be used to restrict the (allowed)
class of interactions through their range and/or through the allowed
strength of the many-body terms. (If one is interested in phase
diagrams one should isolate the one-body energies as adjustable
parameters, namely as chemical potentials.) We will require that
$g(\omega^b) \geq 1$ for $\omega^b$ in $\Omega^r$; note that
$g$ may depend on the diameter of $b$.
To see the connection between interactions and ground states we need
some more detailed notation. For each $b$ in $B$ let $d(b)$ denote
the number of elements of $\Omega_b$, and let $B_0 =
\{b_j\ \vert\ j\in {\bf N}\}$
be some ordering of the countable set $B_0$.
Let $K$ denote the set of all ordered pairs $\{(j,k)\
\vert\ j,k \in {\bf N},\ k \leq d(b_j)\}$. We can use $K$ to label the elements of
$\Omega^r$: for each $(j,k)$ in $K$ let
$\omega_{(j,k)}$ be the $k^{th}$ of the $d(b_j)$ possible
restrictions to $b_j$.
The space $\Omega$, as a subset of the Cartesian product
$A^{{\bf Z}^d},$ is equipped with the usual product topology and Borel
measurable sets, both generated by the (basic) cylinder sets,
$\{C_{(j,k)}\ \vert\ (j,k)\in K\}$, associated with the elements
$\omega_{(j,k)}$ of $\Omega^r$.
Let $F$ be the set of all translation invariant Borel probability
measures on $\Omega$. Now given any measure $m$ in $F$ and interaction $V$
satisfying 2), the energy density in $m$ is just $$
\ \equiv \sum_{(j,k)\in K}V(\omega_{(j,k)})\ m(C_{(j,k)}).\eqno 3)$$
As is well known (see [30,33,4]), if $m_0(V)\in F$ is a ``ground state
for $V$'' in the sense that $m_T(V)[C_{(j,k)}]\to m_0(V)[C_{(j,k)}]$
for all $(j,k)\in K$ as $T\to 0$, where $m_T(V)$ is a Gibbs state for
the interaction $V$ at temperature $T$, then $m_0(V)$ satisfies $$\ = \inf_{m\in F}.\eqno 4)$$
\nd (Note that 4) is necessary but not sufficient for ground states; just
consider the free model, where $V(\omega_{(j,k)}) = 0$ for all
$(j,k)$). It is easy to show [4,32,33] that any ground state for an
interaction $V$ has its support in the set of all ``ground state
configurations'' for $V$, namely the set of configurations defined as
follows. We define the energy $E_b(\omega) = \sum_{b':b'\cap b\ne
\emptyset}V(\omega^{b'})$. The ground state configurations of $V$ are then
those configurations $\omega$ such that for all $b$ in $B$
$$E_b(\omega) = \inf_{\hat \omega\in\Omega}
\{E_b(\hat \omega)\ \vert \ \hat \omega^{b'}=\omega^{b'} \hbox{ for all }
b' \hbox{ such that }b'\cap b = \emptyset\}.
\eqno 5)$$
\nd We remark that the set of all ground state
configurations for a given interaction is compact.
Finally we note that for a generic $V$ the set of all ground state
configurations is uniquely ergodic [24]. This strongly suggests, by
reference to the Gibbs phase rule [31,10], that the set of ground state
configurations of a low temperature pure thermodynamic phase is
uniquely ergodic.
\vskip .2truein
\nd {\bf 3.\ First Method.}
What concerns us in this paper is the degree to which ground states
need be highly ordered. The greatest possible order occurs when
$m_0(V)$ is of the form $m_{\omega}$ defined as follows: take any
periodic element $\omega \in \Omega$, let $m'_{\omega}$ be the
point mass concentrated at $\omega$, and let $m_{\omega}$ be the
(finite) convex combination of point masses that you get by averaging
$m'_{\omega}$ over translations. Aside from degenerate models, a
noninteracting system being an extreme example, there are very few
examples known of interactions which do not have ground states of this
simple periodic type. However, there are interactions $V,$ obtained
from Wang tilings [23,25,26,27], where the support of $m_0(V)$ is the
orbit closure, $\bar O(\omega)$, of a ``quasiperiodic'' configuration
$\omega$, and contains no periodic configurations. This uniquely
ergodic $m_0(V)$ is still highly ordered: for example it is not
strongly mixing and has zero entropy -- as holds for generic
interactions [15,28]. (The spectrum of the translations is presumably
pure point, as is expected for ``real'' quasicrystals.) The main
objective of this paper is to exhibit examples of interactions $V$ for
which the ground state $m_0(V)$ is uniquely ergodic on the ground
state configurations but highly {\bf disordered}; for example weakly
or strongly mixing, and/or with positive entropy.
To produce such interesting ground states one can begin with any
``appropriate'' configuration $\omega$, where by appropriate we mean
that $\bar O(\omega)$ is strictly ergodic; in such a case denote by
$m_{\omega}$ the corresponding unique translation invariant
probability measure on $\bar O(\omega)$. (This agrees with the special
case treated above, namely that of periodic $\omega$.) For example [3]
consider the configuration $\hat \omega$, for dimension $d = 1$ and
with $A =
\{1,-1\}$, obtained as follows. Let $\hat \omega^0 = 0$ and define
$\hat \omega^j$ by the following iterative (``substitution'') process for
$j \geq 0$ -- replace all $0$'s by $001$, and replace all $1$'s by
$11100$ --
$$0\ \to \ 001,\ \ 1\ \to \ 11100.\eqno 6)$$
\nd (So after one iteration we have $\hat \omega^1 = 0$ and $\hat
\omega^2 = 1$, and after the second iteration we have $\hat \omega^3 =
0,\
\hat \omega^4 = 0,\ \hat \omega^5 = 1,
\ \hat \omega^6 = 1,\ \hat \omega^7 = 1,\ \hat \omega^8 = 1,\ \hat \omega^9
= 0,$ and $\hat \omega^{10} = 0$; etcetera.)
We could now define $\hat \omega^j$ for $j < 0$, but in fact this is
unnecessary since the orbit closure under translations of this
``partial'' configuration is already uniquely defined, and has the
above mentioned property that this orbit closure is the support of a
unique translation invariant probability measure, $m_{\hat \omega}$
[3,22].
Now that we have fixed the measure $m_{\hat \omega}$, we need to define
the interaction $V^{\hat \omega}.$ First we prove the following simple
lemma.\vskip .1truein
\nd {\bf Lemma}. If the compact set $X \subset \Omega$ is minimal
invariant under lattice translations and supports the invariant probability
measure $m$, then: $$X = \Bigl\{\omega\in \Omega\
\Bigl\vert\ \{C_{(j,k)}\ \vert\ m(C_{(j,k)}) > 0\} = \{C_{(j,k)}\
\vert\ C_{(j,k)}\cap O(\omega) \ne \emptyset\}\Bigr\}.$$
\nd {\bf Proof}. Fix some point $\omega' \in X$. It follows from minimality
(Prop.~IV.1 in [22]) that $$X = \{\omega\in \Omega\ \vert
\hbox{ for every }(j,k)\in K,\
O(\omega)\cap C_{(j,k)} \ne \emptyset \iff O(\omega')\cap
C_{(j,k)} \ne \emptyset\}.$$ If $\omega \in X$ and for some cylinder set
$C_{(j,k)}$ we have $O(\omega)\cap C_{(j,k)} \ne \emptyset$, then
(Prop.~IV.5 in [22]) we must have $m(C_{(j,k)}) >
0$. Also, if $\omega \in X$ and for some cylinder set $C_{(j,k)}$ we
have $m(C_{(j,k)}) > 0$, then $\omega''
\in C_{(j,k)}$ for some $\omega'' \in X$, and therefore
$O(\omega)\cap C_{(j,k)} \ne \emptyset$. So if $\omega \in X$
then $m(C_{(j,k)}) > 0$ if and only if $O(\omega)\cap C_{(j,k)}
\ne \emptyset$. Finally, if $\omega \notin X$ then there is some
cylinder set $C_{(j,k)}$ disjoint from $X$ such that $\omega\in
C_{(j,k)}$ and so $O(\omega)\cap C_{(j,k)} \ne \emptyset$. But
$m(C_{(j,k)}) = 0$ from the disjointness and so for $\omega\notin
X$ $$\{C_{(j,k)}\ \vert\ m(C_{(j,k)}) > 0\} \ne \{C_{(j,k)}\
\vert\ C_{(j,k)}\cap O(\omega)\ne \emptyset\}.\qed$$
\vskip .1truein
So with the configuration $\hat \omega$ defined above we associate
(following Aubry [2]) the interaction $V^{\hat \omega}$ by:
$V^{\hat \omega}[C_(j,k)] > 0$ if $m(C_{(j,k)}) = 0$, and
$V^{\hat \omega}[C_(j,k)] = 0$ if $m(C_{(j,k)}) > 0$. (This does not
completely define $V^{\hat \omega}$ of course, but we can select the
positive values of the interaction so that the interaction will belong
to any of the allowed spaces of interactions associated with the norms
$\Vert\ \cdot\ \Vert_g$ considered above; in particular, the
interaction can have any prescribed rate of decrease with diameter or
with the number of particles in the many-body terms.) It follows by
the above procedure that the set of ground state configurations of $V^{\hat
\omega}$ is uniquely ergodic, so $m_{\hat \omega}$ is the unique ground
state of $V^{\hat \omega}.$ We note the following features of $m_{\hat
\omega}$ -- it is weakly mixing, and so its spectrum contains only one
eigenvalue, the number 1, with multiplicity 1, the rest of the
spectrum being continuous [3].
We have just constructed an interaction whose ground state is
associated with a certain (rather wild) prescribed configuration,
$\hat \omega$. By this method we can in fact produce even wilder
examples. It is a rather surprising theorem of Jewett, Krieger and Weiss
[11,12,35] that given any invertible ergodic measure preserving
transformation on a Lebesgue measure space, with finite entropy $S$,
there is a measure theoretic isomorphism with a strictly ergodic
symbolic dynamical system, that is with the type of lattice system we
have been using, where $\exp(S) < N(A) \leq \exp(S) + 1$. This is an
existence theorem, which guarantees the existence of $\omega$'s with
$m_{\omega}$'s of great variety: for example strongly mixing, and/or
with nonzero entropy. Explicit examples are constructed in [7,8]. And
therefore the above proves the existence of interactions with a very
wide range of measures as their unique ground states. The interactions
have as short a range as desired (in the sense of rate of fall-off),
but of course the method does require the use of many-body energies of
all orders. We summarize the above in a theorem. \vskip .1truein
\nd {\bf Theorem 1}. Given any representation of ${{\bf Z}^d}$ by
invertible ergodic measure preserving transformations on a Lebesgue
measure space, with finite entropy, and any norm $\Vert\ \cdot\
\Vert_g$, there is a short range lattice gas model
with interaction $V$ satisfying $\Vert V\Vert_g < \infty$, for which
the set of ground state configurations is uniquely ergodic, and for
which the ground state dynamical system is measure theoretically
isomorphic to the given system.
\vskip .1truein
\nd {\bf Remarks}. The method used above for constructing an
interaction $V^{\omega}$ associated with a given configuration
$\omega$ was first defined by Aubry for the general case where the
orbit closure $\bar O(\omega)$ is minimal [2]. It is important to
emphasize at this point why it is necessary to make further
restrictions on $\omega$, such as the unique ergodicity of
$\bar O(\omega)$.
If one starts with any minimal configuration $\omega$ and uses the
above method to construct an interaction $V^{\omega}$, it follows
easily from 3) and 4) that $\bar O(\omega)$ contains the support of
any zero temperature, translation invariant limit of finite
temperature Gibbs states of the model. If $\omega$ is uniquely ergodic
(but not necessarily minimal), {\bf every} configuration in $\bar
O(\omega)$ (including $\omega$ of course) reproduces the unique
probability measure $m_{\omega}$ in the sense of Birkhoff's pointwise
ergodic theorem; that is, if $\chi$ is the characteristic function for
any cylinder set, and $T^j$ represents translation by $j\in {\bf Z}^d$,
then as $N \to \infty$: $$1/(2N+1)^d
\sum_{\vert j\vert\le N}\chi(T^j\omega') \to
\int_{\bar O(\omega)} \chi\, dm_{\omega}.\eqno 7)$$
\nd (uniformly in $\omega'$) for $\omega'$ in $\bar O(\omega)$
(Thm.~3.5 in [5]). On the other hand, if one uses an $\omega$ which
is minimal but not uniquely ergodic (see section 10 in [18] for an
example), then first of all one loses control of the zero temperature
limits since there are now infinitely many invariant probability
measures, all with the same minimal set as support. Also the
configuration $\omega$ that one started with need not be well
associated with {\bf any} of these measures; in the example from [18]
noted above, some averages of the type in 7) do not even exist for
$\omega' = \omega$. Therefore if one wants to construct an
interaction for which a given configuration $\omega$ is not only a
ground state configuration in the sense of the definition 7), but is
also associated in some real sense with low temperature Gibbs states,
then one must assume more than minimality for $\omega$. We assume
unique ergodicity, in part because it is sufficient for this purpose
but also because as noted at the end of section 2 it seems appropriate
for pure thermodynamic phases.
\vskip .2truein
\nd {\bf 4.\ Second Method.}
Two of the shortcomings of the above result (namely the use of
many-body energies of all orders, and the requirement that the
interaction not be strictly finite range), can be avoided by a
different method described in this section.
First we need to outline two techniques for defining interesting symbolic
dynamical systems. In both cases we define the dynamical system
$X$ as the set of all points $x = \{x^{(j^1,\cdots, j^d)}\
\vert\ (j^1,\cdots, j^d) \in {\bf Z}^d\} \in A^{{\bf Z}^d}$ for which all
``blocks'' $\{x^{(j^1,\cdots, j^d)}\ \vert\ J_1\le j^1\le K_1, \cdots,
J_d\le j^d\le K_d\}$ in $x$ satisfy certain restrictions.
For the class of one dimensional dynamical systems usually associated
with the term ``substitution'', one begins with a set of
``substitution rules'', that is, for each element $a \in A$ one has a
finite sequence $\{a_1, \cdots ,a_k\}$, where $a_j \in A$ and $k$ (the
so-called length of the rule) may depend on $a$. (We used such rules
in 6) to define half of the configuration $\hat \omega$.) Given any finite
sequence $B$ of elements of $A$, we define $D(B)$ as the finite
sequence obtained by replacing each of the elements of $B$ using its
rule. Next define $ V_0 = A,\ V_{n+1} =
\bigcup_{B\in V_n}D(B)$ and $\ V = \bigcup_{n\ge0}V_n$. Then $X$ is the set
of all two sided sequences $x$ for which every subblock of $x$ is a
subblock of an element of $V$. Finally, given a set of substitution
rules defining the set $X$, we say $X$ has ``unique derivation'' if
for every $x\in X$ there is a unique $y\in X$ (unique up to
translation) such that $x$ is obtained from $y$ when the substitution
rules are used to replace the elements of $y$. (Note that
there is no need to keep track of the absolute coordinates $x^j$ of
our two-sided sequences $x$ under substitution, as was done in section
3. This difference is an example of the change in perspective between
focusing on a configuration and on a dynamical system, as discussed in
section 5 below.) For example, the dynamical system associated with
the rules 6) has unique derivation. An example of a dynamical system
without unique derivation is the one with the rules $0\to 010,\ 1\to
101$. See [17,22].
The second technique produces interesting {\bf two} dimensional
dynamical systems $X$, and is associated with the term ``tiling''.
Here one begins by specifying two subsets $K_h$ and $K_v$
of $A\times A$. Both subsets are assumed to satisfy the
condition:
\vskip .1truein
\noindent {\bf i)} If $(a,b),\ (a,c)\hbox{ and }(d,c)\hbox{ are in the
set, so is }(d,b).$
\vskip .1truein
\noindent We then define the dynamical system $X$ as the
subset of $A^{{\bf Z}^2}$ such that all blocks in $X$ of the form
$\{x^{(j^1,j^2)}\ \vert\ J\le j^1\le J+1\}$ are in $K_h$ and all
blocks in $X$ of the form $\{x^{(j^1,j^2)}\ \vert\ J\le j^2\le J+1\}$
are in $K_v$.
We now show how this dynamical system is related to tilings of the
plane. Think of each element of $A$ as a unit square centered over a
point of ${\bf Z}^2$. These unit squares, henceforth called
``tiles'', have four edges (called ``left'', ``right'', ``top'' and
``bottom''), and these edges will be assigned ``colors''. We will be
restricting arrangements of the tiles in the plane by requiring that
they may abut only if the overlapping edges have the same color. If we
think of $K_h$ and $K_v$ as a list of the pairs of tiles that may abut
horizontally and vertically, then we can define colors for the edges
of the tiles by the following prescription. For tile $a$ we define the
color of: the right edge to be $\{(c,b)\in K_h\ \vert\ (a,b)\in
K_h\}$, the left edge to be $\{(c,b)\in K_h\ \vert\ (c,a)\in K_h\}$,
the top edge to be $\{(c,b)\in K_v\ \vert\ (a,b)\in K_v\}$, and the
bottom edge to be $\{(c,b)\in K_v\ \vert\ (c,a)\in K_v\}$. Condition
{\bf i)} ensures that the sets of pairs of tiles used to define colors
are pairwise disjoint, so that the colors are well defined. (Although
we never need to use the colors, we defined them to show that a
definition of colored tiles is possible for which the allowed tilings
-- that is, those tilings in which each pair of abutting edges have
the same color -- satisfy the restrictions of $K_h$ and $K_v$.)
To summarize, the above allows us to interpret $X$, henceforth called
a (two dimensional) tiling dynamical system, as the set of all tilings
of the plane, by the tiles in $A$, such that abutting edges always have
the same color. (For completeness we note that the above two
techniques, for substitution dynamical systems and for tiling
dynamical systems, can both be generalized to other dimensions
[17,5].)
By a recent result of Mozes (Theorem 6.4 in [17]), given any one
dimensional substitution dynamical system with unique derivation and
with substitution rules all of length at least 2, one can build a two
dimensional tiling dynamical system which is measure theoretically
isomorphic to the product of the one dimensional substitution
dynamical system with itself. The one dimensional substitution
dynamical system associated with the rules 6) is easily seen to
satisfy the two hypotheses.
Now given any tiling dynamical system as defined above, assuming $X$
is nonempty (which is automatic in the application below) it is easy
to define a two dimensional lattice gas model, with {\bf nearest
neighbor two-body interaction}, for which the ground state dynamical
system is measure theoretically isomorphic to the tiling dynamical
system [23,25,26,27]. Combining this with the above application of the
theorem of Mozes using 6), and general facts about the product of
weakly mixing dynamical systems (Prop.~4.6 and Thm.~4.30 in [5]) we
have the following example.
\vskip .1truein
\nd {\bf Example}. There is a two dimensional
classical lattice gas model with nearest neighbor two-body
interaction, such that the set of ground state configurations is
uniquely ergodic, and the operators representing translations have
no eigenvalues other than 1.
\vskip .1truein
\nd {\bf Remarks}. The use of the rules 6) is of course just one
example of a general method for producing interesting ground states.
There are some natural shortcomings of this second method also. First
of all, any one dimensional substitution dynamical system has zero
entropy [22] and is not strongly mixing [3,22]. While it is possible
that the above method can be extended to yield two dimensional models
with strongly mixing translations, in fact it is impossible to have a
finite range interaction and nonzero entropy in the ground state, as
we see in the following theorem proven with Jacek Mi\c ekisz. (The
two notions ``topological entropy'' and ``measure theoretic entropy''
are known to coincide for strictly ergodic dynamical systems [19], and
also coincide with the physical entropy for ground states [1], so we
have used the simple term ``entropy''.)
\vskip .1truein
\nd {\bf Theorem 2}. In any dimension $d$, if
a lattice gas model has an interaction of strictly finite range and a
uniquely ergodic set of ground state configurations, then the ground
state has zero entropy.
\vskip .1truein
\nd Proof. For simplicity we only consider dimension $d = 2$. Assume the
(topological) entropy of the ground state $m$ is $\alpha > 0$, and the
range of the interaction is $R$. Define $S_N$ as the square set of
$N^2$ sites in the lattice centered at the origin, ${\bf C}_N$ as the
set of cylinder sets based in $S_N$, and ${\bf D}_{N,R}$ as the set of
cylinder sets based in $S_{N+R}/S_N$. From the definition of
(topological) entropy, the number of $C \in {\bf C}_N$ for which $m(C)
> 0$ is $\alpha^N$ asymptotically in $N \to \infty$. The number of
elements of ${\bf D}_{N,R}$ is $N(A)^{4R\surd N}$. Let $\omega$ be
any configuration in the support of $m$, so $\omega
\in C'$ for some $C'
\in {\bf C}_N$ for which $m(C') > 0$. Let $ D$ be the
element of ${\bf D}_{N,R}$ to which $\omega$ belongs. Consider the
translations of $\omega$ by fewer than $N$ units along both axes;
there are fewer than $4N$ of them, and so there are fewer than $4N$
cylinders $C \in {\bf C}_N$ to which these translations of $\omega$
belong. From the above, for fixed $N$ large enough there exists a
cylinder $C'' \in {\bf C}_N$ for which $m(C''\cap D) > 0$ and which
contains none of these translations of $\omega$. We now construct the
configuration $\omega''$ as follows. For each translation $T$ such
that $T(\omega)
\in C'\cap D$ we change $\omega$ at $N^2$ sites so that $T(\omega) \in
C''\cap D$. It is easy to see that this new configuration is also a
ground state configuration. But if we let $\chi$ be the
characteristic function for $C''\cap D$, it follows from Prop.~IV.5 in
[22] that the left hand side of 7) has a different limit if $\omega' =
\omega$ than if $\omega' = \omega''$, which is a contradiction. Therefore
$\alpha = 0$.\qed
\vskip .2truein
\nd {\bf 5.\ Summary.} There were two goals of this article. The first
was to show how the mathematics of strictly ergodic dynamical systems
can play a powerful role in mathematical physics, for example in
the crystal problem.
The second goal was to exhibit statistical mechanical models of
traditional type with unprercedented levels of disorder in their
ground state. In particular we described two methods for producing
models with short range, translation invariant interactions with
disordered ground states; the methods are constructive in some
circumstances (when enough is known about the desired ground state, as
in the examples discussed), and nonconstructive when dependent on
existence theorems such as that of Jewett, Krieger and Weiss.
The models have ground states exhibiting unusual spectral
disorder (the translation operators have continuous spectrum) even
with nearest neighbor two-body interactions; interactions having
positive entropy ground states are constructed but are proven to
require that the interaction not be strictly finite range.
Examples of interactions whose unique ground state has positive
entropy (using [8] in our second method, for instance) are of some
special interest. It is well known that models can have ground
states of positive entropy, as in the Ising antiferromagnet on the
triangular lattice. However in such a model the entropy is produced
by degeneracy, while that is not so clear in our examples; if there is
only one invariant probability measure on the ground state
configurations of an interaction, in some strong sense the model is
nondegenerate. So the traditional connection between positive entropy
and degeneracy needs clarification. It need not be associated with
{\bf local} nonuniqueness (that is, with a freedom to pass from one
ground state configuration to another by any of many strictly local
changes) as in the antiferromagnet example.
There is one further aspect which is clarified by the use of uniquely
ergodic dynamical systems. In statistical mechanics one
tends to think of a ground state as associated with a configuration,
but this is not always justifiable even when it is uniquely ergodic.
The ground state is really an (invariant) probability measure, and
only effectively reduces to a configuration when the ground state is
periodic as described in section 3. The example of the tiling models
mentioned in section 1 is apposite. For tiling models, if one ignores
the existence of fault lines or planes by only considering the support
of the uniquely ergodic ground state measure, one finds that all
configurations in the support have precisely the same finite patterns
in them (and with the same frequencies). Therefore in some physical
sense they are indistinguishable; knowledge of any finite region of a
configuration cannot characterize the configuration. Therefore it
seems more reasonable to think of the ground state of a pure phase as
the (unique, uniquely ergodic) probability measure rather than to try
to associate a configuration with the ground state. An historical
reason to associate configurations with ground states, and with other
low temperature ordered states, is the existence in some models of
translation noninvariant Gibbs states which are in some sense
perturbations of some fixed configuration, as in the nearest neighbor
Ising antiferromagnet on the square lattice. It should be remembered
however that, in principal at least, a low temperature phase need not
be distinguished by having nonunique (in particular, translation
noninvariant) Gibbs states; it is conceivable that, say, a tiling
model has a low temperature phase described by a unique Gibbs state.
The positive temperature behavior of tiling models and others as
discussed in this paper should be determined. (For tiling models see
[13,14,16].) The fact that entropy can enter these models in a new
manner suggests that the usual method for estimating low temperature
Gibbs states, using Peierls contours [31], may need essential
revision.
\vskip .5truein
\nd{\bf Acknowledgements.} It is a pleasure to acknowledge useful
conversations with Hans Koch, Jacek Mi\c ekisz and
Aernout van Enter. We also thank W. Veech and P. Diaconis for
referring us to the work of Grillenberger and Mozes, respectively.
Theorem 2 was proven jointly with Jacek Mi\c ekisz while attending
a workshop at the Santa Fe Institute, the support of which is
gratefully acknowledged.
\hfil\eject
\baselineskip=12pt \nopagenumbers
\pageno=13
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\end