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\input amssym.tex
\vsize = 23.5 truecm
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\hoffset = .2truein
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\nopagenumbers
\def\ski{\smallskip \noindent}
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%centerline {Invited Talk}
%\centerline {XXVII International Workshop on Condensed Matter Theories}
%\centerline {Toulouse, France, September 15-20, 2003}%

%\centerline {to appear in: Condensed Matter Theories, Vol. {\bf 19}, Nova
%Science Publisher, N.Y.}
%\centerline {e-mail: kuerten@ap.univie.ac.at}
%\pacs{PACS numbers: 07.05.Mh, 02.20.-a, 05.30.Fk, 07.05.Pj}
\topinsert
\vskip 3.2 truecm
\endinsert
\centerline{\bigbfont MULTISTABILITY AND FRACTAL PROPERTIES}
\vskip 8truept
\centerline{\bigbfont OF HAMILTONIAN LATTICE MODELS}
\vskip 20 truept
\centerline{\bigifont Karl E. K\"urten}
%$^*$}
\vskip 8 truept
\centerline{Institute for Experimental Physics, University of Vienna} 
\centerline {5, Boltzmanngasse, A-1090 Vienna, Austria}
\vskip 12 truept
\centerline{\bigifont Christian Krattenthaler}
%$^*$}
\vskip 8 truept
\centerline{Institut Girard Desargues,
Universit{\'e} Claude Ber\-nard Lyon-I,} 
\centerline {21, avenue Claude Ber\-nard,
F-69622 Villeurbanne Cedex, France}
\vskip 1.8 truecm


\centerline{\bf 1.  INTRODUCTION}
\vskip 12 truept
Numerous condensed matter systems are effectively discrete by nature  because 
the relevant length scales are of the order of the interparticle distance. 
Such systems are described by a Hamiltonian that is discrete in space,
 while their time evolution
is considered as continuous. Their remarkable behavior, exemplified
in models of charge-density waves, magnetic spirals, disordered crystals, magnetic
multilayers 
or adsorbed monolayers, stems from a competition between two or more forces
that leads to  
multistable spatially modulated 
structures [1--5]. One other prominent example 
is modeling proton dynamics in hydrogen
bonded chains 
[6].
Spatial disorder is 
associated 
with a highly complex energy landscape in configuration space, 
where the number of locally stable configurations typically increases 
exponentially with the size of the system. The simplest model 
 system can be 
envisioned as a one-dimensional chain of 
$N$ particles with nearest-neighbour interactions, each particle, which is 
considered as a classical  spin, also subject
to an external multi-well potential field.
Specifying the displacement of particle $i$ at 
time $t$ 
from its lattice site by the variable $x_i$, and assuming periodic
boundary conditions, 
the total energy of the system is given by the $N$-particle Hamiltonian
$$
{\bf H} (x_1,x_2,\dots,x_N) =
\sum_{i=1}^N 
\left({1 \over 2} {\dot x_i}^2 + E_I(x_{i-1},x_i,x_{i+1}) + U(x_i)\right).
\eqno(1)
$$
The first term is the kinetic energy, the second  term 
defines the inter-site energy, specified by nearest-neighbour
interactions, while  
the last term is the potential energy due to an 
external on-site potential. 

A standard harmonic inter-site potential is given by
$$
E_I(x_{i-1},x_i,x_{i+1})=
{1 \over 4} (x_i - x_{i-1})^2 + 
{1 \over 4} (x_i - x_{i+1})^2 , 
$$                                                               
while the on-site potential is specified by the $\Phi^4$ 
double-well potential 
$$
U(x_i) = {\alpha \over 4}(1 -   x_i^2 )^2 ,
\eqno(2)
$$                                                              
depicted in Fig.~1. 
It is
often used for the description of displacive and order-disorder transitions.
Here, the control parameter $\alpha$ specifies the height of the barrier
$U(0) = {\alpha \over {4 }}$ 
which separates the two symmetrical 
sinks of the two potential wells.
%The equations of motion are given by the $N$ 
%coupled non-linear wave equations
%$$
%{{\ddot x}_i} - 
%(x_{i-1} - 2 x_i + x_{i+1}) - \alpha x_i (1 - x_i^2) = 0
%   \qquad i=1,\dots,N
%\eqno(3)
%$$                                                              
\vskip 10 truept \noindent
\vbox{
\input psfig.sty
\centerline
{%\hskip10mm
\psfig{figure=v4.ps,height=6truecm,width=12truecm,angle=270}}
\centerline{{\bf Figure 1.} 
Symmetric $\Phi^4$ double-well potential for $\alpha=8$.}}
%\vskip 10truept \noindent
\vskip 8truept \noindent
Based on the $\Phi^4$ potential, 
the total energy 
Eq.~(4) 
takes the form
$$
{\bf E} =
\sum_{i=1}^N 
\left({1 \over 2} {\dot x_i}^2 + 
{1 \over 4} (x_i - x_{i-1})^2 + 
{1 \over 4} (x_i - x_{i+1})^2  +{\alpha \over 4}(1 -   x_i^2 )^2 \right). 
\eqno(3)
$$
The ground state with zero energy is obtained,
when  all particles are 
placed in one of the two sinks of
the double-well potential.
According to the variational principle, stationary configurations obey 
the set of $N$ 
non-linear 
coupled equations
$$
-x_{i-1} + 2 x_i - x_{i+1} - \alpha x_i (1 - x_i^2) = 0 , \qquad i=1,\dots,N.
\eqno(4)
$$                                                              
\vskip 8truept \noindent 
However, solutions ${\bf x}^*=(x_1^*,x_2^*,\dots,x_N^*)$ specify a local 
minimum only if 
the eigenvalues of the triangular Hessian  
$
{\bf \nabla}^2 {\bf H}({\bf x}) _{|{\bf x = x^*}}=({\partial^2 {\bf H}({\bf x})
\over {\partial x_i {\partial x_j}}})  _{|{\bf x = x^*}}
$
are all positive.
Since the diagonal entries  are $2 + U''(x_i^*)$ and the two 
off-diagonal entries take the value $-1$,
Gershgorin's disk theorem provides a  sufficient stability condition 
$U''(x_i^*) > 0$, where all particles occupy 
the convex part of the potential well. For the double well potential this
condition is satisfied for $x_i^2 > {1 \over 3}$.

There are two macroscopic quantities of interest. 
First, the total energy,
which takes the simple form, reminiscent of the virial theorem in the
continuous counterpart of the nonlinear Klein-Gordon equation,
$$
E = {\alpha \over {4}} \sum_{i=1}^N ( 1 - x_i^4),
\eqno(5)
$$
which is obtained
after insertion of the equilibrium condition Eq.~(4).
Second, the average displacement
$$
M = {1 \over N} \sum_{i=1}^N x_i ,
\eqno(6)
$$
reminiscent of the total magnetization of a classical spin
chain of interacting magnetic particles, a macroscopic variable, which is
often experimentally accessible. In this case the intersite energy $E_I$ specifies
the exchange interaction between two neighbouring particles, 
while the on-site potential plays the role of the
uniaxial anisotropy energy with strength parameter $\alpha$ 
[7].
\vskip 28truept \noindent
\centerline{\bf 2. FEW PARTICLE CASE}
\vskip 12truept
\noindent                     
Let us first consider the situation for $N=2$ particles, where the energy
takes the symmetric form
$$
E(x_1,x_2) = (x_1-x_2)^2 + {\alpha \over 4} 
\left((1-x_1^2)^2 +(1-x_2^2)^2 \right) . $$
The set of coupled equations
$$
\eqalignno{-x_2 + 2 x_1 - x_2 - \alpha x_1 (1 - x_1^2) &= 0,\cr  
-x_1 + 2 x_2 - x_1 - \alpha x_2 (1 - x_2^2) &= 0 .}
$$
has $9$ real valued solutions representing stationary configurations.
Fig.~2 
shows the energy landscape $E(x_1,x_2)$ for $\alpha=8$ characterized by 
one  relative maximum  (0,0) 
and four local minima
$\pm (1,1)$, and
$\pm \left(\sqrt{1 - {4 / \alpha}},-\sqrt{1 - {4 / \alpha}}\right)$.
The first two minima represent the two degenerate ground states with
energy $E=0$, while 
the latter represent two energetically higher lying
antiferromagnetic states with energy per particle
$
E/2=  2 \left(1- {2\over \alpha}\right).
$
%and
%Kratt: Habe "they" eingefuegt.
%they are stable only for $\alpha > 6$.
Note that these four local minima are separated by four saddle
points with energy per particle
$
E/2 =  { {(2+\alpha)^2}/(8 \alpha}) .
$

The saddle points act as energy barriers, 
whose height increases linearly with
increasing control parameter $\alpha$. It is important to note that 
the height of the barriers is of the order of $\alpha$, while
the energy values of the local
minima are all of the order of one. 
Accordingly, the energy difference between these states is much smaller
than the energy required to overcome the barrier in order to  switch to
another state of equilibrium.
\vskip 8 truept                                           
\vbox{
\input psfig.sty
\centerline
{%\hskip10mm
\psfig{figure=twodim.ps,height=6truecm,width=12truecm,angle=270}}
\centerline{{\bf Figure 2.} Energy landscape $E(x_1,x_2)$ for  $\alpha=8$.}
}
\vskip 10truept \noindent
The same arguments hold for larger values of $N$, where we will
show in 
Section~3 that the energy landscape     
exhibits 
an extremely complicated multi-valley structure with an exponentially
increasing number 
of local minima and
saddle points that also allows the appearance of structural disorder.

\noindent
For $N=3$ we have a "soliton" solution
$$
{\bf x}^* = \{ x_1,x_2,x_1 \},
$$
with
$$
x_1 =
%\sqrt {{\alpha \over {c}}}
\left[{  {2 \over 3} - {1 \over \alpha} + {2 \over 3}
 \left(1 - {3 \over \alpha}\right)^{1 \over 2} ~
{\rm cos}\, {{\phi + \pi} \over 3} } \right]^{1/2}   
%Kratt: Das waeren doch ein biszchen wenig Abstaende.
%\hbox{and} 
\quad \hbox{and}\quad 
x_2 = x_1 (1 - \alpha + c x_1^2),
$$
and with
$$
\cos \phi = {
{ 1 - {9 \over {2 \alpha}} + {27\over \alpha^3}} \over
{(1 - {3 \over \alpha})^{3 \over 2}   } }.
$$
\smallskip \noindent
This solution represents a local minimum  only 
for $\alpha >  3 + \sqrt{9 + 6 \sqrt{3}} \approx 7.40367$
with the expansion  
$
x_1 =
1 - {2 \over \alpha} 
%- {3 \over \alpha^2} - {11 \over \alpha^3} 
%-{105 \over{2 \alpha^4 }}
   -\cdots 
%-\sqrt {\alpha \over c}
\quad \hbox{and} \quad
x_2 =
%{\sqrt {\alpha \over c}
-1 + {1 \over \alpha} 
%- {2 \over \alpha^3} 
-\cdots 
%-{21 \over{2 \alpha^4 }}.
$, respectively.
\vskip 28truept \noindent
\centerline{\bf 3. EXACT ENUMERATION OF ALL LOCALLY STABLE STRUCTURES}
\vskip 12truept
Let us now further concentrate on large values of $\alpha$,
where the on-site potential is sufficiently strong. 
One might ask: which kind of landscape- 
or domain structures can evolve due to
the
competition between the exchange-
and the on-site energy?
Depending on the number of particles
$N$, how many distinct 
non-equivalent 
structures can there exist?
When the control parameter $\alpha$ is varied, 
the energy balance is changed such that
a rearrangement of the  energy landscape can take place. 
Stationary
configurations of the Hamiltonian (1)  
are described by
the $N$ non-linear coupled equations
$$
-x_{i-1} + 2 x_i - x_{i+1} - \alpha x_i (1 - x_i^2) = 0 , \qquad i=1,\dots,N.
\eqno(7)
$$
In order to determine all possible locally stable spatial
structures  
we first consider the system 
for large values of $\alpha$. In the large $\alpha$-limit,
Eq.~(7) 
reduces to 
$$
 x_i (1 - x_i^2) = 0 , \qquad i=1,\dots,N .
\eqno(8)
$$
Ignoring the solutions $x_i=0$, which correspond to saddle point configurations,
this coupled set of equations has  exactly $2^N$ distinct solutions  
consisting of $N$-dimensional binary strings
$$
{\bf x}^*=(x_1^*,x_2^*,\dots,x_N^*)
\eqno(9)
$$  
with components $x_i^* \in 
 \{-1,1\}$. 
Provided
that $\alpha$ is sufficiently large, the implicit function theorem
guarantees the existence of a solution of Eq.~(7)
in a neighbourhood of any of these binary strings, all of these
solutions being distinct.
(Experimentally, the critical $\alpha$ depends weakly 
on the particle number $N$, 
%Kratt: Das ist aber gar nicht, was man sagen will.
%however it turns out that numerical solutions of Eq.~(7) exist
however, numerical calculations suggest that in fact $\bf all$ $2^N$ locally 
stable solutions of Eq.~(7) exist
in a neighbourhood of any of these binary strings 
provided that $\alpha \approx >8$.) 
If these are
expressed in terms of a
power series expansion about the
asymptotic strings $\bf x^*$, then
the first order correction term 
$$
\Delta x_i^{(1)} = {x_{i-1}^* - 2 x_i^* + x_{i+1}^*  \over {2 \alpha}} 
\eqno(10)
$$
is proportional
to $1/\alpha$. 
The particular form of the first- and the higher order correction terms
then implies that $|x_i| \leq 1$ rigorously holds. 
It should be noted however that, 
depending on a specific
solution,  the stability range 
can vary widely.

Since the particles reside on a 
circle, we will consider configurations that only differ by a rotation
$(x_i  \rightarrow x_{s+i})_{1\le i\le N}$ 
(for some fixed $s$ with $1\le s\le N$) or by a
reflection $(x_{i} \rightarrow x_{s-i})_{1\le i\le N}$ 
(for some fixed $s$ with $1\le s\le N$)
as equivalent. Together, these rotations and reflections form the
dihedral group ${\cal D}_N$.
Since for large $\alpha$, the stable solutions of (7)
can be identified with the $2^N$ 
strings of 1s and $-1$s of length $N$
(for example, the ``soliton" solution for $N=3$ presented in Section~2
would be identified with the string $(-1,1,-1)$),
the number of inequivalent solutions will be equal to the number
of orbits of these (circular) strings under the action of
the dihedral group ${\cal D}_N$. In more colloquial terms, we want to
count the number
$F_D(N)$ of different necklaces one can make out of $N$ beads,
each of them either coloured white or black. It follows from 
Redfield-P\'olya theory (see for example [8]), 
that the number of these orbits (or necklaces), 
and thus the number of inequivalent
stable solutions to (7), is
%$$
%F_C(N) = 
% {1 \over {N}} \sum_{d | N} \varphi(N/d)~ 2^{N/d}  
%\eqno(7)
%$$
$$
F_D(N) = 
 {1 \over {2 N}} \sum_{d | N} 
\varphi(d)\, 2^{N/d} +  
\cases {{3 \over 4} 2^{N/2} & for $N$ even, \cr
    2^{(N-1)/2}&for $N$ odd,}
\eqno(11)
$$
where $\varphi(x)$ is the Euler totient 
function.
For $N=1,2,3,\dots$ this is the sequence
$\{2,3,4,6,8,13,18,30,46,78,126,224,\dots\}$.
Since $\varphi(1)=1$, the number
$F_D(N)$ is asymptotically equal to $2^{N-1}/N$ 
as $N$ tends to $\infty$.
%The presence of random features or structural disorder does not
%necessarily in the Barkhausen effect reflects the
%general need for statistical methods and concepts to treat magnetization
%processes and domain structures.

In order to enumerate the orbits of strings that are invariant
under reflections and those which are not separately, we also count
the number $F_C(N)$ of different necklaces made out of $N$ beads, 
each of them coloured white or black, {\it which are
embedded in the plane}, that is, the orbits of strings of 1s and $-1$s
of length $N$ under the action of the cyclic group ${\cal C}_N$ of
rotations $(x_i  \rightarrow x_{s+i})_{1\le i\le N}$. Redfield--P\'olya
theory again shows that
$$F_C(N)={ {1}\over {N}}
\sum _{d\mid n} ^{}\varphi(d)2^{N/d}.$$
The first few terms of this sequence are
$\{2, 3, 4, 6, 8, 14, 20, 36, 60, 108, 188, 352,\dots\}$.

As is easy to see,
the number of orbits {\it under the action of the dihedral group ${\cal D}_N$}
which are {\it not\/} invariant under reflection
is then $F_C(N)-F_D(N)$, while the number of the orbits
under the action of the dihedral group ${\cal D}_N$ which are 
is $2F_D(N)-F_C(N)$.

The spectrum of the total magnetizations $M$ as well as the energy
spectrum, consisting of $8267$ distinct values,
both reminiscent of Cantor-set structures, are depicted in 
Fig.~3 and Fig.~4,
respectively.
\vskip 10truept
\vbox{
\input psfig.sty
\centerline{\psfig{figure=toufin2.ps,height=3truecm,width=16truecm,angle=270}}
\noindent
\centerline{{\bf Figure 3.}
Spectrum of the total magnetizations Eq.~(6) for $N=15$ and $\alpha = 8$.}}
\noindent
\vskip 6truept
\vbox{
\input psfig.sty
\centerline{\psfig{figure=toufin3.ps,height=3truecm,width=16truecm,angle=270}}
\noindent
\centerline{{\bf Figure 4.}  Energy spectrum Eq.~(5) for $N=15$ and $\alpha = 8$.}}
\vskip 9 truept \noindent
Note that a specific domain structure corresponds to the system being 
trapped in one of these
local energy minima. 
If thermal fluctuations are neglected 
and the energy barriers
separating this minimum from neighbouring ones are large enough, the system
will indefinitely remain in such a metastable state. 
However, a slight change of the strength of the 
on-site potential $U(x)$ can easily destabilize a specific domain structure
such that it is sufficient that a local minimum of the
energy landscape is transformed 
into a saddle point and 
the system can evolve towards some other 
metastable configuration.  
\vskip 28truept
\centerline{\bf 4. EXACT ENUMERATION OF ALL DISPLACEMENTS}
\vskip 12truept
In order to determine the distribution or the "spectrum" of all possible displacements 
$\vert x_i\vert$, $i=1,2,\dots,N$,  as 
a function of the particle number $N$, 
we exploit the fact that two-dimensional maps allow 
us to represent any stationary configuration
of a Hamiltonian of the form (1) by a trajectory of a dynamical system.
Therefore, the mathematical situation is {\it identical} to that of
temporal evolution, although the problem has been formulated
in terms of spatial arrangements.
Introducing the auxiliary variable 
$y_{n+1} = x_n,$
Eq.~(7) takes the form of a two-dimensional iterative map in discrete time,
$$
  \left( \matrix { x_{n+1} \cr
                  y_{n+1} }  \right)
={\bf F}(x_n,y_n) =
  \left( \matrix {-y_n + 2 x_n - \alpha x_n(1-x_n^2)  \cr
                 x_n      }   \right)
% \qquad n = 0,1,\dots
\eqno(12)
$$
Due to the 
periodic boundary conditions, we have to determine all $F_D(N)$ inequivalent
periodic orbits of the map equation. These can be easily calculated, 
once the fixed
points 
$$
\left( \matrix { x^*  \cr
                 y^*     }   \right)
={\bf F}^{(N)} (x^*,y^*)
\eqno(13)
$$
of the $N$-th iterate of the 
map are known. Since periodic orbits of the map equation
can be shown to be unstable with respect to iterating the map 
[9], we
solve the nonlinear set Eq.~(13) with a standard Newton-Raphson
 procedure, where 
the asymptotic strings Eq.~(9)  serve as initial conditions.

We first calculate all possible displacements as the 
length $N$ of the chain is varied, where we first concentrate on $N$ odd. 
Fig.~5 shows the displacement spectrum for odd $N=3,5,7,9$.
Experimentally, with increasing number of particles, the
spectrum  
evolves by consecutive tripling and quadrupling of
spectral substructures. 
It obviously grows in a quasi self-similar manner, 
where structures replicate themselves on successively smaller
scales with respect to their statistical properties.
Triplets
split into a quadruplet surrounded by two triplets,
while quadruplets split into four quadruplets.
Hence, it seems that the spectrum grows according to
$$
 S(N) = 2 S(N-2) + 4^{ {N-3} \over 2} .
\eqno(14)$$
\vskip 6truept \noindent
\input psfig.sty
\centerline{\hskip-3mm
\psfig{figure=toufrac1.ps,height=3truecm,width=12truecm,angle=270}}
\noindent
\vskip 6truept \noindent
\input psfig.sty
\centerline{\hskip-3mm
\psfig{figure=toufrac2.ps,height=3truecm,width=12truecm,angle=270}}
\noindent
\vskip 6truept
\input psfig.sty
\centerline{%\hskip10mm
\psfig{figure=toufrac3.ps,height=3truecm,width=12truecm,angle=270}}
\noindent
\vskip 6truept
\vbox{
\input psfig.sty
\centerline{%\hskip10mm
\psfig{figure=toufrac4.ps,height=3truecm,width=12truecm,angle=270}}
\noindent
\centerline{{\bf Figure 5.}  Spectrum of the displacements for 
$N=3$, $N=5$, $N=7$,}
\centerline{and $N = 9$ (from above) for $\alpha = 8$ .}
}
\vskip 9truept \noindent     
For $N=3,5,7$, and 
$N=9$,  
we have $3,10,36$, and $136$ 
distinct values, respectively. 
An estimate for the fractal dimension 
can be obtained by using a box
counting algorithm, and it yields the
value $d_f=0.58$.
%It should be noted, 
%however, that the spectrum is not self-similar in the strict sense. 
%For $N=1,2,\dots$ the total number of
%distinct displacements is given by the sequence
%$$
%\{1,2,3,6,10,20,36,72,136,272,528,1056,2080,4160,8256,16512,32896,65792,
%\dots\}.
%$$

A rigorous verification of the recurrence (14) is equivalent to
rigorously determine the numbers $S(N)$ of distinct displacements.
We shall be able to do this under the hypothesis, which
is confirmed by our experiments, that displacements in one orbit are 
distinct from the displacement in another orbit, and that 
the number of distinct displacements in an orbit is equal to
the minimal period $s$ under rotation of the corresponding binary string 
${\bf x}^*=(x_1^*,x_2^*,\dots,x_N^*)$ if the orbit is not invariant
under reflection, and equal to $(s+1)/2$ otherwise. 
Thus, one way to determine $S(N)$ is to calculate the sum
${{1} \over{2}}\sum _{{\bf x}^*} ^{}w({\bf x}^*)$, where the sum is over all 
orbits ${\bf x}^*$, and $w({\bf x}^*)=s$, respectively
$w({\bf x}^*)=(s+1)/2$, depending on the two cases described above.
(The factor ${1\over 2}$ results from 
the invariance $x_i \rightarrow -x_i$ in Eq.~(7)). 
There is a similar weighted sum for even $N$.

It is possible to do this calculation directly. There is, however, an
elegant way around this. 
Consider the set of all binary strings constructed in the following
manner: For any orbit, let ${\bf x}^*=(x_1^*,x_2^*,\dots,x_N^*)$ 
be the corresponding binary
string, with weight $w({\bf x}^*)=w$, say. List the strings
$$
\eqalign{
&(x_1^*,x_2^*,\dots,x_{N-1}^*,x_N^*)\cr
&(x_2^*,x_3^*,\dots,x_{N}^*,x_1^*)\cr
&(x_3^*,x_4^*,\dots,x_{1}^*,x_2^*)\cr
&\dots\dots\dots\dots\dots\dots\dots\dots\cr
&(x_w^*,x_{w+1}^*,\dots,x_{w-2}^*,x_{w-1}^*).
}$$
Finally form the union of all these strings over all orbits.
The upshot is that this
set of strings is a set of representatives of orbits of 
strings of 1s and $-1$s of length $N$ under the action of the group
generated by the two operations of
reversal of strings $r:(x_i^*)\to(x_{N-i+1}^*)$, 
and the operation
of simultaneous exchange of $1$s and $-1$s $e:(x_i^*)\to(-x_{i}^*)$.
This group consists only of 4 elements, the identical operation
$(x_i^*)\to(x_{i}^*)$, the reversal $r$, the exchange $e$, and the
composition $re:(x_i^*)\to(-x_{N-i+1}^*)$ of reversal and exchange. Thus,
with the help of 
Redfield-P\'olya theory, we arrive at the explicit expression
$$
S(N) = 
 2^{N-2}+ \cases{ 2^{(N-2)/2} & for $N$ even, \cr
         2^{(N-3)/2}&for $N$ odd,}
\eqno(15)
$$
which had been empirically found earlier in [7] and confirms indeed the
empirically observed recurrence Eq.~(14).

%\vfill \eject
\vskip 28truept 
\centerline{\bf 4. DISCUSSION}
\vskip 10truept
We have shown that 
the energy surface of our model consists of exponentially 
many locally stable minima separated
by large barriers specified by unstable saddle points.
With the aid of Redfield-P\'olya theory we give the exact number
of inequivalent locally stable configurations as well
as the exact number of distinct values of the displacement
spectrum. 
Both quantities increase exponentially with increasing particle
number $N$. 
As $N$ tends to $\infty$  they are 
asymptotically 
$2^{N-1}/N$ and 
$2^{N-2}$ , respectively.
The spectrum, which grows exponentially in a quasi self-similar manner 
is shown to be a fractal
with fractal dimension $d_f = 1.55$. 

%From the physical point of view these results have numerous interpretations.
%The model could for instance describe regular and irregular motion of a proton
%in an external double-well potential created by two neighbouring
%heavy ions in the adiabatic approximation, where the 
%vibrations of the heavy ions are neglected [4]. The two-well oscillations 
%would describe the proton escape from one potential well to the other, 
%being either regular or chaotic.  
The model with slight modifications 
can also describe the formation of magnetic 
structures in  physical novel systems such as antiferromagneticly
coupled  magnetic 
multilayers, where modulated structures arise
due to the competition between the exchange energy, the anisotropic
on-site energy and an additional external magnetic field [7]. 
Here the fractal structure is intimately connected with the
experimentally observed
Barkhausen jumps which 
exhibit fractal properties
associated
with random Cantor dust of a fractal dimension $d_f$ depending on the
strength of the external magnetic field [10].
\vskip 28 truept
\centerline{\bf ACKNOWLEDGMENTS}
\vskip 12 truept
The first author thanks S. Charpentier, M. Olivier and M. Martin for 
helpful computer assistance and 
F.V. Kusmartsev for various useful discussions. 
\vskip 28 truept
\noindent
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\end 
