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\title{The Global Minimum of Energy Is Not Always a Sum of Local Minima - a
Note
on Frustration}
\author{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.}
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\begin{document}
\baselineskip=26pt
\maketitle
{\bf Abstract.} A classical lattice gas model with translation-invariant
finite
range competing interactions, for which there does not exist an equivalent
translation-invariant finite range nonfrustrated potential, is constructed.
The
construction uses the structure of nonperiodic ground state configurations
of
the model. In fact, the model does not have any periodic ground state
configurations. However, its ground state - a translation-invariant
probability
measure supported by ground state configurations - is unique.
KEY WORDS: Frustration; m-potential; nonperiodic ground states; tilings.
\eject
\section{Introduction}
Low temperature behavior of systems of many interacting particles results
from
the competition between energy and entropy, i.e., the minimization of the
free
energy. At zero temperature this reduces to the minimization of the
energy density. Configurations of a system which minimize its energy
density are
called ground state configurations. One of the important problems of
statistical
mechanics is to find ground state configurations for given interactions
between
particles. If we can find a configuration such that potential energies of
all
interactions between particles are minimal then we can conclude that it is
a
ground state configuration. It is then said that such a model is not
frustrated.
Otherwise, we may rearrange potentials and construct an equivalent
Hamiltonian
which may not be frustrated and which will enable us to find ground state
configurations. Here I present a classical lattice gas model with
translation-invariant finite range competing interactions for which there
does
not exist an equivalent translation-invariant finite range nonfrustrated
potential. In other words: the global minimum of energy is not the sum of
its
minima attained locally in space. More precisely, one cannot minimize the
energy density of
interacting particles by minimizing their energy in a finite box and all
its translates, no matter
how large is the box.
\section{Classical Lattice Gas Models, Frustration, and M-Potentials}
A classical latice gas model is a system in which every site of a lattice
$Z^{d}
$ can be occupied by one of $n$ different particles. An infinite lattice
configuration is an assignment of particles to lattice sites, that is an
element
of $\Omega=\{1,...,n\}^{Z^{d}}$. Particles can interact through many-body
potentials. A {\em potential} $\Phi$ is a collection of real valued
functions
$\Phi_{\Lambda}$ on configuration spaces
$\Omega_{\Lambda}=\{1,...,n\}^{\Lambda}$ for all finite $\Lambda \subset
Z^{d}$.
Here we assume $\Phi$ to have finite range, that is $\Phi_{\Lambda} = 0$ if
the
diameter of $\Lambda$ is large enough, and be translation-invariant. The
formal Hamiltonian can be
then written as $$H= \sum_{\Lambda}\Phi_{\Lambda}.$$
Two configurations $X,Y \in \Omega$ are said to be {\em equal at infinity},
$X
\sim Y$, if there exists a finite $\Lambda \subset Z^{d}$ such that $X = Y$
outside $\Lambda.$ The relative Hamiltonian is defined by
$$H(X,Y)=\sum_{\Lambda}(\Phi_{\Lambda}(X)-\Phi_{\Lambda}(Y)) \; \;for \;
\;X \sim
Y.$$ $X \in \Omega$ is a {\em ground state configuration} of $H$ if
$$H(Y,X) \geq 0 \; \; for \; \; any \; \; Y \sim X.$$ For any potential the
set
of ground state configurations is nonempty but it may not contain any
periodic
configurations \cite{rad1,rad2,ram}. We will be concerned here with
nonperiodic
ground state configurations which have uniformly defined frequencies for
all
finite patterns. By definition the orbit closure of such a ground state
configuration supports
a unique strictly ergodic translation-invariant measure called a ground
state
which is a zero temperature limit of a low temperature Gibbs state (an
infinite
volume grand canonical probability distribution). If we can find a
configuration that minimizes all $\Phi_{\Lambda}$, then it is necessarily a
ground state configuration and we call such potential nonfrustrated or an
m-potential \cite{slaw1,slaw2}. Formally, a potential $\Phi$ is an
{\em m-potential} if there exists a configuration $X$ such that
$$\Phi_{\Lambda}(X)=min_{Y}\Phi_{\Lambda}(Y) \; \; for \; \; any \; \;
finite \;
\; \Lambda.$$
Otherwise, we may try to rearrange interactions to obtain an equivalent
m-potential. Two potentials are defined to be {\em equivalent} if they
yield the
same relative Hamiltonian and therefore have the same ground state
configurations and the same Gibbs states. It is best illustrated by an
example
of the antiferromagnetic nearest neighbor spin $1/2$ model on the
triangular lattice. The formal Hamiltonian can be written as follows:
$$H = \sum_{i,j}\sigma_{i}\sigma_{j},$$
where $\sigma_{i}, \sigma_{j}= \pm 1$ and $i$ and $j$ are nearest neighbor
sites on the triangular lattice. When you look at an elementary triangle it
is
easy to see that at least one pair of spins does not minimize its
interaction.
Two spins allign themselves in opposite directions and then the third one
can
minimize only one of the two remaining interactions. This choice is a
source of
frustration \cite{tou} (see also another approach to frustration
\cite{and,mie}). However,
we may construct the following equivalent potential:
$$\phi_{\triangle} =
1/2(\sigma_{i}\sigma_{j} + \sigma_{j}\sigma_{k} + \sigma_{k}\sigma_{i}),$$
where
$i$, $j$, and $k$ are vertices of an elementary triangle $\triangle$ and
$\Phi_{\Lambda}=0$ otherwise. Now, there are ground state configurations
minimizing every $\Phi_{\Lambda}.$ Three spins on every elementary triangle
still face choices but they act collectively and therefore are not
frustrated.
In the following section I construct an example of a lattice gas model
with nearest neighbor translation-invariant frustrated interactions for
which
there does not exist an equivalent finite range translation-invariant
m-potential. The main problem of proving the impossibility of an
m-potential is
that a grouping of interactions in big plaquettes, like in the above
example, is
not the only way of constructing an equivalent m-potential. To construct it
one may also
use an information about a global structure of excitations. In some models
just
grouping is clearly impossible because energy can be lower locally than
that of
a ground state configuration and you can pay for it arbitrarily far away,
yet
one can still construct an equivalent m-potential. One of the easiest
examples is a
one-dimensional Ising model with the following interactions: the energy
of $+-$ neighbors is equal to $-1$, the energy of $-+$ neighbors is $2$,
and
otherwise the energy is zero. There are arbitrarily long line segments with
the energy equal to $-1$.
Nevertheless, the above potential is equivalent to an m-potential with the
energy of $-+$ neighbors equal to $1$ and zero otherwise, or
$-1/4(\sigma_{i}\sigma_{i+1}-1)$ using
spin variables.
\section{An Intrinsically Frustrated Model}
The model is based on Robinson's
tiles \cite{rob,pat}. There is a family of 56 square-like tiles such that
using
an infinite number of copies of each of them one can tile the plane only in a
nonperiodic fashion. This can be translated into a lattice gas model in the
following way first introduced by Radin \cite{rad1,rad2,ram}. Every site of
the
square lattice can be occupied by one of the 56 different particles-tiles.
Two
nearest neighbor particles which do not ``match'' contribute positive
energy,
say 2; otherwise, the energy is zero. Such a model obviously does not have
periodic ground state configurations. There are uncountably many ground
state configurations but
only one translation-invariant ground state measure supported by them.
There is a one-to-one
correspondence between ground state configurations in the support of this
measure and Robinson's
nonperiodic tilings. Low temperature behavior of this model was
investigated in \cite{mie1,mie2,mie3}.
I describe now slightly modified Robinson tiles \cite{mye}. There are seven
basic tiles represented
symbolically in Fig.1. The rest of them can be obtained by rotations and
reflections. The first tile
on the left is called a cross; the rest are called arms. All tiles are
furnished with one of the four
parity markings
shown in Fig.2.
The crosses can be combined with the parity marking at the lower left in
Fig.2. Vertical arms (the direction of long arrows) can be
combined with the marking at the lower left and horizontal arms with
the marking at the upper left. All tiles may be combined with the remaining
marking. Two tiles ``match'' if arrow head meets arrow tail. Let us observe
that if the plane is tiled with tiles with such markings then these parity
markings must
alternate both horizontally and vertically in the manner shown in Fig.2.
Let me now describe the main features of the Robinson's nonperiodic
tilings.
I will concentrate on the lattice positions of crosses denoted by
$\lfloor,
\lceil, \rfloor, \rceil$, where directions of line segments correspond to
double
arrows in Fig.1. Every odd-odd position on the $Z^{2}$ lattice is occupied
by
these tiles in relative orientations as in Fig.3. They form the periodic
configuration with the period 4. Then in the center of each ``square'' one
has
to put again a cross such that the previous pattern reproduces but this
time
with the period 8. Continuing this procedure infinitely many times we
obtain a
nonperiodic configuration. It has built in periodic configurations of
period
$2^{n}$, $n\geq 2$ on sublattices of $Z^{2}$ as shown in Fig.4.
Now I will modify the above model a little bit introducing another level
of markings which are optional, that it is to say they can be present or
absent in appropriate tiles.
Every cross can be equipped with one of the two markings shown at the left
in Fig.5. The orientation
of a marking at the top should be the same as the orientation of double
arrrows of its cross and it comes in either red or green color. The second
marking comes only in
red color. Arms at the left in Fig.1 can be furnished with red or green
lines shown in the middle
column in Fig.5. Finally, arms at the top of Fig.1 can be equipped with
either a red marking
at the upper right in Fig.5 or a marking at the lower right in Fig.5 with
green-red, red-green, or green-green sides. Now, two tiles match if there
are no
broken lines and adjacent colors are the same. In the corresponding
classical
lattice gas model in addition to two-body nearest neighbor interactions I
will
introduce a chemical potential equal to $1$ for green crosses and having a
negative value $\tau$ for red crosses and zero value for uncolored crosses
and all arms.
\newtheorem{ground}{Proposition}
\newtheorem{frus}{Theorem}
\begin{ground}
For $\tau>-1$ a unique ground state measure of the modified model is the
same
as of the original Robinson model.
\end{ground}
{\em Proof:} Let a broken bond be a segment on the dual lattice separating
two nearest neighbor
particles with a positive interaction energy. I will prove that in any
ground state
configuration broken bonds are absent. Let us observe that the matching
rules are such that on
any $2^{n}Z^{2}$ sublattice, $n \geq 1$, in any region without broken bonds,
crosses should be
oriented like in Fig. 3 except possibly on a sublattice boundary. The
lower bound of the energy of interacting particles in any such region is
obtained if crosses on the
boundary of every sublattice are red and all other crosses are not colored
and we take into account
only the negative energy of chemical potentials of red crosses. For any
broken bond there can be at
most two red crosses hence the energy of a configuration is at least
proportional to the total length
of broken bonds (a variation of a Peierls condition is satisfied). This
shows that in any ground
state configuration broken bonds are absent. Among configurations without
broken bonds,
configurations without any colored particles (Robinsons's original
configurations) have the minimal
energy density (equal to zero) and are therefore the only ground state
configurations.$\Box$
Obviously, our interactions do not constitute an
m-potential. Moreover, it is impossible to construct a
translation-invariant
finite range m-potential by grouping interactions in big plaquettes like it
was done in the antiferromagnetic example. One may locate colored crosses
on
vertices of a square with a size $2^{n}$ therefore decreasing energy
locally and
paying for it arbitrarily far away [see Fig.6]. Now I will prove that for
some
$\tau$ an equivalent translation-invariant finite range m-potential does
not
actually exist.
\begin{frus}
The above described model for $-1 < \tau <4/5$ does not have an equivalent
translation-invariant finite range m-potential.
\end{frus}
{\em Proof:} Let us assume otherwise and let its range be smaller than
$2^{n}.$
Let us consider three local excitations from a Robinson ground state
configuration shown in Fig.6, where squares have size $2^{n+1}.$ Equating
relative Hamiltonians for the original interaction and a hypothetical
equivalent m-potential we obtain:
\begin{equation}
\tau+3=a_{r}+b_{rg}+c_{g}+d_{g}+e_{g}+f_{g}+g_{g}+h_{gr},
\end{equation}
\begin{equation}
2\tau+2=a_{r}+b_{rg}+c_{g}+d_{g}+e_{g}+f_{gr}+g_{r}+h_{r}+i_{r},
\end{equation}
\begin{equation}
2\tau+2=a_{r}+b_{r}+c_{r}+d_{rg}+e_{g}+f_{g}+g_{g}+h_{gr}+i_{r},
\end{equation}
where on the right hand sides we have nonnegative contributions to energy
due
to a hypothetical m-potential and coming from regions labelled in the
upper left corners of the squares in Fig.6; subscripts correspond to
configurations of optional markings with $r$ denoting red and $g$ denoting
green.
Now, set $\tau =-1+\delta /2.$ From (2) we obtain $a_{r} \leq \delta$
and $b_{rg}+c_{g}+d_{g}+e_{g} \leq \delta$, and from (3)
$f_{g}+g_{g}+h_{gr} \leq \delta.$ Then it follows from (1) that
$a_{r} \geq 2-3/2\delta$ which contradicts $a_{r} \leq \delta$ if
$\delta<4/5.$ This contradiction rules out the existence of an equivalent
translation-invariant finite range m-potential. $\Box$
\section{Conclusions}
A classical lattice gas model with
translation-invariant nearest neighbor competing interactions is
constructed.
Its unique translation-invariant ground state measure is supported by
nonperiodic ground state
configurations. There are local excitations in the model such that
the energy is locally lower than that of a ground state configuration and
one
pays for it arbitrarily far away. This shows that by grouping interactions
in big
plaquettes, like in the antiferromagnetic model on the triangular lattice,
one
cannot construct an equivalent finite range m-potential. More generally it
is proved that such a
potential actually does not exist. The model is therefore intrinsically
frustrated.
Let us note that in the antiferromagnetic model a spin on an
elementary triangle is frustrated because it faces a choice of direction.
Its both choices can be present in a ground state configuration making
therefore a ground state highly degenerate. In our example a particle may
choose a local minimum of energy and then it appears that this does not
lead to
a ground state configuration.\\
\vspace{5mm}
{\bf Acknowledgments.} I
would like to thank Alan Sokal and Roberto Fernandez for an inspiration, and
Jean Bricmont for helpful discussions. Bourse de recherche UCL/FDS is
gratefully acknowledged for the financial support.
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