Vanishing N\'eel Ordering of SU($n$) Heisenberg Model in Three Dimensions
Noboru Fukushima
The SU($n$) Heisenberg model represented by exchange operators is studied by means of high-temperature series expansion in three dimensions, where $n$ is an arbitrary positive integer. % The spin-spin correlation function and its Fourier transform $S(\mathbf{q})$ is derived up to $O[(\beta J)^{10}]$ with $\beta J$ being the nearest-neighbor antiferromagnetic exchange in units of temperature. % The temperature dependence of $S(\mathbf{q})$ and next-nearest-neighbor spin-spin correlation in the large $n$ cases show that dominant correlation deviates from $\mathbf{q}=(\pi,\pi,\pi)$ at low temperature, which is qualitatively similar to that of this model in one dimension. % The N\'eel temperature of SU(2) case is precisely estimated by analyzing the divergence of $S(\pi,\pi,\pi)$. Then, we generalize $n$ of SU($n$) to a continuous variable and gradually increases from $n=2$. We conclude that the N\'eel ordering disappears for $n>2$.
3dSUn.tex
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\begin{document}

\title{
Vanishing N\'eel Ordering of SU($n$) Heisenberg Model in Three Dimensions
}


\author{Noboru Fukushima}
\email{fukushima@lusi.uni-sb.de}
\affiliation{
Department of Physics, University of the Saarland,
66041 Saarbr\"ucken, Germany}



\begin{abstract}
The SU($n$) Heisenberg model represented by exchange operators is studied by
means of high-temperature series expansion in three dimensions, where
$n$ is an arbitrary positive integer.
%
The spin-spin correlation function and its Fourier transform $S(\mathbf{q})$
is derived up to $O[(\beta J)^{10}]$ with $\beta J$ being the
nearest-neighbor antiferromagnetic exchange in units of temperature.
%
The temperature dependence of $S(\mathbf{q})$ and next-nearest-neighbor
spin-spin correlation in the large $n$ cases show that dominant correlation
deviates from $\mathbf{q}=(\pi,\pi,\pi)$ at low temperature, which is
qualitatively similar to that of this model in one dimension.
%
The N\'eel temperature of SU(2) case is precisely estimated by analyzing
the divergence of $S(\pi,\pi,\pi)$.  Then, we generalize $n$ of SU($n$) to a
continuous variable and gradually increases from $n=2$.  We conclude
that the N\'eel ordering disappears for $n>2$.
\end{abstract}

\pacs{
75.10.Jm %Quantized spin models
75.40.Cx %Static properties ( heat capacities, etc.)
75.50.Ee %Antiferromagnetics
}


\maketitle

\section{Introduction}

It is known that properties of quantum spin systems tend to approach
those of their corresponding classical spin systems as the spin magnitude
increases.
%
However, this is not necessarily the case for a sequence of models in
which the number of multipolar-interaction terms increases as the spin
magnitude increases.  Consequently the higher-spin systems may have
stronger quantum effects in this case.
%
In other words, such additional terms can break a classical
correspondence down even in high dimensions.
%
These ``large-spin-magnitude'' systems mentioned above include systems
in which one unit has more than two degrees of freedom, such as
orbitally degenerate systems\cite{Kugel82, FK,KF}.  In such
systems, many coupling constants appear in general.  However,
experimental information about multipolar couplings is limited.  As a
starting point to explore such systems, understanding of one of the
extreme limits must be useful.  Therefore in this paper, we investigate
an SU($n$) symmetric case
\cite{Uimin70,Sutherland75,Klumper99,Affleck86,Kawakami92,Batchelor03,
Yamashita98,Frischmuth99,Fukushima02,Fukushima03,
lima,vdb,RVBsu4,fermiMF,shen,Penc03,
Ohkawa85,Shiina97,chen72}
in three dimensions, where $n$ is the total
number of internal degrees of freedom, by means of high-temperature
series expansion (HTSE).


We consider a simple cubic lattice. Let each site take one of the $n$
colors denoted by $|\alpha\rangle$ with $\alpha = 1, 2, \cdots, n$.  Using
the Hubbard operator $X^{\alpha\beta} := |\alpha\rangle\langle\beta|$,
the exchange operator is expressed as
%
\begin{equation}
 P_{\mathbf{i},\mathbf{j}} := \sum_{\alpha=1}^n  \sum_{\beta=1}^n
X_\mathbf{i}^{\alpha\beta} X_\mathbf{j}^{\beta\alpha}.
\end{equation}
%
Colors of sites $\mathbf{i}$ and $\mathbf{j}$ are exchanged when
$P_{\mathbf{i},\mathbf{j}}$ is applied.
%
Then, an SU($n$) symmetric Hamiltonian reads
\begin{equation}
 {\cal H} :=  J \sum_{\langle \mathbf{i},\mathbf{j}\rangle} P_{\mathbf{i},\mathbf{j}},
\label{hamil}
\end{equation}
%
where the summation is taken over all the nearest neighbor pairs.
We consider the antiferromagnetic case, $J>0$.
Let us show the relations with spin operators explicitly for
some of the special cases below.

(i) When $n=2$, this model is reduced to the ordinary Heisenberg model
with $s=1/2$ by relation
$2P_{\mathbf{i},\mathbf{j}}-1=4 \bm{s}_\mathbf{i}\cdot \bm{s}_\mathbf{j}$.

(ii) The SU(3) case corresponds to $s=1$,
competing quadratic and biquadratic exchange interaction,\cite{PaPanicolaou88}
\begin{equation}
1+P_{\mathbf{i},\mathbf{j}} =
\bm{s}_\mathbf{i}\cdot
\bm{s}_\mathbf{j}+(\bm{s}_\mathbf{i}\cdot \bm{s}_\mathbf{j})^2.
\label{eq:su3rep}
\end{equation}


(iii) The SU(4) Heisenberg model is
related to spin 3/2 systems but more often discussed in the context of
orbital- and spin-degenerate systems\cite{Yamashita98,Frischmuth99,
Fukushima02,Fukushima03,
lima,vdb,RVBsu4,fermiMF,shen,Penc03,
Ohkawa85,Shiina97}.
In particular, the model in three dimensions has been used
as an effective model of CeB$_6$ to explain magnetic-field dependence of
the transition temperature of an antiferro-orbital ordering.
\cite{Ohkawa85,Shiina97}
The four local states
can be represented by
$|+)|+]$, $|+)|-]$, $|-)|+]$, $|-)|-]$,
where $|\pm)$ and $|\pm]$ represent an orbital state and a spin
state, respectively. The pseudo-spin operators are defined
by $t^z|\pm)=\pm\frac12|\pm)$, $t^\pm|\mp)=|\pm)$,
$s^z|\pm]=\pm\frac12|\pm]$, $s^\pm|\mp]=|\pm]$, and
the exchange operator for $n=4$ is rewritten as
\begin{equation}
P_{\mathbf{i},\mathbf{j}} = \bm{t}_\mathbf{i}
\cdot\bm{t}_\mathbf{j} + \bm{s}_\mathbf{i}\cdot \bm{s}_\mathbf{j}
  +4 (\bm{t}_\mathbf{i}\cdot\bm{t}_\mathbf{j})  (\bm{s}_\mathbf{i}\cdot \bm{s}_\mathbf{j})
 +\frac{1}{4}.
\end{equation}
The Pauli matrices $\bm{\tau}=2\bm{t}$ and $\bm{\sigma} =2\bm{s}$
may simplify this expression, {\em i.e.},
\begin{equation}
4 P_{\mathbf{i},\mathbf{j}}
= \bm{\tau}_\mathbf{i}\cdot\bm{\tau}_\mathbf{j}
+ \bm{\sigma}_\mathbf{i}\cdot \bm{\sigma}_\mathbf{j}
  + (\bm{\tau}_\mathbf{i}\cdot\bm{\tau}_\mathbf{j})  (\bm{\sigma}_\mathbf{i}\cdot \bm{\sigma}_\mathbf{j})
 +1.
\label{paulirep}
\end{equation}


Note that there is a different representation of SU($n$) Heisenberg model
studied in detail using Quantum Monte Carlo method
(QMC)\cite{Harada02}.  However, the QMC for the Hamiltonian
(\ref{hamil}) suffers from minus sign problems in more than one
dimensions \cite{Harada02, Frischmuth99}


If the number of competing order parameters is large and frustration
exists, the transition temperature can greatly be reduced from the
mean-field value even in three dimensions. That will be the case with the
Hamiltonian (\ref{hamil}).
%
First, it is isotropic with respect to $n^2-1$ independent interacting
components.  Furthermore, it contains frustration as most clearly seen
in Eq.~(\ref{paulirep}) of the SU(4) case.  Each of the 15 components,
$\tau^\alpha$, $\sigma^\beta$, $\tau^\gamma \sigma^\delta$, attempts an
antiparallel alignment.  However, it cannot be attained simultaneously,
{\em e.g.}, simultaneous N\'{e}el states of $\tau^z$ and $\sigma^z$ produce a
ferromagnetic alignment of the product $\tau^z \sigma^z$.  Such
frustration is not so explicit in the SU(3) case in Eq.~(\ref{eq:su3rep})
as in the SU(4) case, yet appearance of the minus-sign problem in the
QMC indicates that similar frustration lies in it.\cite{Harada02}

This frustration should become stronger as $n$ increases.  One can see
in one dimension how the system finds a compromise against this
frustration.  The SU($n$) Heisenberg model in one dimension can be
exactly solved.\cite{Uimin70,Sutherland75,Klumper99, Affleck86,
Kawakami92,Batchelor03} An important point is that the ground state has a
quasi--$n$-site periodicity.  The SU(4) case is numerically studied in
more detail; ground-state properties by the density matrix
renormalization group\cite{Yamashita98}, and thermodynamic properties by
the QMC\cite{Frischmuth99} and the HTSE\cite{Fukushima02,Fukushima03}.
%
The real-space spin-spin correlation function as a function of spin-spin
distance has a positive value every four sites.  Its Fourier transform
has a outstanding cusp at $q=\pi/2$ and a small cusp at $q=\pi$.
Regarding the temperature dependence, as temperature decreases
correlation with two-site periodicity develops, and then at lower
temperature another correlation occurs with $n$-site periodicity.
\cite{Fukushima02}

Such peculiar correlation in one dimension could suggest an exotic
ordering in higher dimensions. However, not much is known about the
antiferromagnetic SU($n$) Heisenberg model in three dimensions.
%
In fact, its ferromagnetic variant is studied in Ref.~\onlinecite{chen72}
by the HTSE for the uniform susceptibility that needs less effort to be
calculated than susceptibility of other wave-numbers.
%
To our knowledge, our calculation in this paper is the first HTSE aiming at
antiferromagnetic exchange of this model.
%
In this study, we investigate the model systematically by changing
parameter $n$ of SU($n$), and report some unique features of spatial
correlation at finite temperature.  Namely, in Sec.~\ref{sec:corfunc} we
show that the behavior of correlation functions is similar to that in one
dimension, and in Sec.~\ref{sec:neeltem} that the N\'eel temperature
disappears as $n$ increases from $n=2$.




\section{Temperature dependence of correlation functions}
\label{sec:corfunc}

What should be calculated here is $\langle P_{\mathbf{i},\mathbf{j}} \rangle$,
which is related to a correlation function.
For example, when $n=4$, fifteen components in Eq.~(\ref{paulirep})
contribute equally, and thus
$
\langle \tau_\mathbf{i}^\alpha \tau_\mathbf{j}^\alpha\rangle
=
\langle \sigma_\mathbf{i}^\beta \sigma_\mathbf{j}^\beta\rangle
=\langle \tau_\mathbf{i}^\gamma \sigma_\mathbf{i}^\delta
         \tau_\mathbf{j}^\gamma \sigma_\mathbf{j}^\delta \rangle
=
\langle 4 P_{\mathbf{i},\mathbf{j}} -1 \rangle /15 $.
%
For general $n$, we can define correlation function between sites
$\mathbf{i}$ and $\mathbf{j}$ as
\begin{equation}
S_{\mathbf{i}-\mathbf{j}} := \langle X_\mathbf{i}^{\alpha\beta} X_\mathbf{j}^{\beta\alpha} \rangle=
\frac{1}{n^2-1} \left( \langle  P_{\mathbf{i},\mathbf{j}} \rangle -\frac{1}{n} \right),
\end{equation}
for $\mathbf{i}\neq \mathbf{j}, \alpha \neq \beta$, which does not
depend on $\alpha$ nor $\beta$ because of the SU($n$) symmetry.
Its Fourier transform is denoted by $S(\mathbf{q})$.


The high-temperature expansion
is performed by expanding
the Boltzmann factor $e^{-\beta {\cal H}}$
in $\beta$.
%
In practice, the series coefficients in the thermodynamic limit
are exactly obtained by a linked-cluster expansion.\cite{Domb3}
%
To obtain the series expansion of
$\langle P_{i,j} \rangle$ up to $O[(\beta J)^M]$,
we need to calculate ${\rm Tr}[({\cal H}_{\rm L})^m]$ and
${\rm Tr}[P_{i,j}({\cal H}_{\rm L})^m]$ for $0\le m\le M$,
where ${\cal H}_{\rm L}$ is the Hamiltonian in a linked cluster.
In calculating the traces, we use
a property of the permutation operator, and
calculation of the traces is reduced to a combinatorial problem
to count the number of circular permutations in a product of permutations.
\cite{Handscomb64,chen72,Fukushima03}
%
Here, an important point of our analysis is that the series coefficients are
obtained as polynomials of $n$, for example,
\begin{eqnarray}
S(\pi,\pi,\pi)&=&
\frac{1}{{n}} + \frac{6}{{ n^2}}(\beta J) + \frac{36}{{ n^3}}(\beta J)^2
\nonumber \\
&&+ \frac{\left( 216 - 22\,{ n^2} \right)}{{ n^4}}(\beta J)^3 +\ldots.
\end{eqnarray}
Namely, the order of the series for every $n$ is the same.
%
We have
obtained the series for $S_{\mathbf{i}-\mathbf{j}}$ up to $O[(\beta
J)^{9}]$ for arbitrary $\mathbf{i}- \mathbf{j}$, and consequently
$S(\mathbf{q})$ up to $O[(\beta J)^{9}]$ for arbitrary $\mathbf{q}$.  As a
special interest, $S(\pi,\pi,\pi)$ is obtained up to $O[(\beta
J)^{10}]$. The series are extrapolated using the Pad\'{e} approximation (PA).

First of all, in order to see the temperature-dependent nature of the
spatial correlation, we analyze the \textit{next}-nearest-neighbor
correlation function $S_{110}$ as we have done in one dimension in
Ref.~\onlinecite{Fukushima02}.  Figure \ref{fig:nncor} shows the
results.  Both the axes are scaled so that the high-temperature limit of
every $n$ matches.
%
Here, we have simultaneously plotted extrapolation from several
different choices of the PA, and the difference between them
approximately represents an error of the extrapolation.

\begin{figure}\begin{center}
\includegraphics[width= 8cm]{110.eps}
\end{center}
\caption{\label{fig:nncor}
Temperature dependence of the next-nearest-neighbor correlation function,
$S_{\mathbf{i}-\mathbf{j}}$ with $\mathbf{i}-\mathbf{j}=(110)$.
}
\end{figure}


The lowest order of the series of $S_{\mathbf{i}-\mathbf{j}}$ has
a N\'eel-order-type correlation in any $n$.
That is, the series of $S_{i_xi_yi_z}$ starts with
$O[(\beta J)^{|i_x|+|i_y|+|i_z|}]$ with sign $(-1)^{|i_x|+|i_y|+|i_z|}$.
%
Therefore, $S_{110}>0$ at high temperature.
%
However, as antiferromagnetic correlation of each interacting component
becomes larger, each short-range order disturbs another because of the
frustration.  The change of the sign of $S_{110}$ suggests that the
correlation acquire a longer period at low temperature.  For $n\ge6$, it
is clear that $S_{110}$ changes the sign at a low temperature that
increases with $n$ in this scale.  For smaller $n$, the relevant
temperature range is below the converged region, and it is difficult to
conclude from this data.


Such a change of correlation at low temperature also appears in the
Fourier transform of the correlation function.
%
A naive extrapolation of the series for $S(\mathbf{q})$ shows bad
convergence for several $\mathbf{q}$.
%
This is probably because information at position $\mathbf{x}$
is lacking when $\cos(\mathbf{x}\cdot \mathbf{q})\simeq 0$.
%
In order to avoid it, we extrapolate the series of a complex function
$ \sum_{\mathbf{x}\ge0}
\langle X_j^{\alpha\beta} X_{j+x}^{\beta\alpha} \rangle
e^{-i\mathbf{x}\cdot\mathbf{q}}  w(\mathbf{x})$,
and after that take the real part of the extrapolated function.
Here, the summation is taken for $x\ge0$, $y\ge0$, $z\ge0$, and
$w(\mathbf{x})$ is two to the power of the number of nonzeros in
$x,y,z$.
This method also makes use of its imaginary part, and the convergence
becomes better than the naive extrapolation.

As we have seen in the
next-nearest-neighbor correlation function, the analysis becomes easier
as $n$ increases.  Therefore, we show $S(\mathbf{q})$ of the SU(16) case
in Fig.~\ref{fig:sq} for a diagonal direction in the $\mathbf{q}$-space,
$ S(q,q,q) $.  Here, its high temperature limit, which does not
depend on $\mathbf{q}$, is subtracted.
%
As temperature decreases, antiferromagnetic correlation develops and
$S(\pi,\pi,\pi)$ increases as the curve at $T=2J$ shows.  However, with
further decrease of temperature, $S(\pi,\pi,\pi)$ starts decreasing and
$S(\mathbf{q})$ with other $\mathbf{q}$ increases.  It is not very clear
from Fig.~\ref{fig:sq} if the maximum of $S(\mathbf{q})$ starts moving
from $(\pi,\pi,\pi)$. Hence, let us show another quantity.  Note that if
the second derivative of $S(\mathbf{q})$ at $\mathbf{q}=(\pi,\pi,\pi)$
changes sign, the position of the maximum clearly moves.
Its temperature dependence for the SU(16) model is plotted in
Fig.~\ref{fig:ddsqn16}.
It shows a change of the sign around $T\sim J$. In fact, this
temperature of changing sign seems to hardly depend on $n$ of SU($n$).
The result above suggests that the N\'eel order disappear at least in
the SU($n$) model with large $n$. Namely, order with wave number
$\mathbf{q}\neq(\pi,\pi,\pi)$, or disorder, should appear.
Then, the next question is, at which $n$ the N\'eel order disappears.


\begin{figure}
\begin{center}\includegraphics[  width=8cm
]{n16sq}\end{center}
\caption{\label{fig:sq}
The Fourier transform of the correlation function
as a difference from its high-temperature limit,
$ S(q,q,q) - S(\mathbf{q})|_{T\rightarrow\infty}$,
for the SU(16) model.
}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width= 8cm]{ddsqn16.eps}
\end{center}
\caption{\label{fig:ddsqn16}
Temperature dependence of
${{\rm d}^2 S(q,q,q)}/{{\rm d}q^2}$ at $q=\pi$
for the SU(16) model by two different PAs.
}
\end{figure}




\section{N\'eel Temperature}
\label{sec:neeltem}

It is known that the SU(2) Heisenberg model has the N\'eel order
at low temperature. Therefore in this section, we gradually
increases $n$ of SU($n$) from $n=2$.
%
As a preparation for that,
we find a reliable way to estimate a transition temperature first,
using the SU(2) model.
%
The N\'eel temperature $T_{\rm N}$ can be characterized by divergence of
$S(\pi,\pi,\pi)$, namely, by a singularity of $S(\pi,\pi,\pi)$ as a
function of $\beta J$.  In order to analyze such a singularity, we use
so-called D-log-Pad\'e approximation (DLPA), {\em i.e.}, the PA for the
logarithmic derivative.
%
In using the DLPA, transformation of the
expansion variable may improve the convergence of extrapolation.
We choose a transformation\cite{Domb13,Pan99}
\begin{equation}
\beta J=\frac{x}{1-{ a}^2 x^2},
\label{eq:trans}
\end{equation}
where $a$ is an adjustable parameter that improves convergence.  In
fact, the singularity closest to the origin in the original series is
near the imaginary axis, while $T_{\rm N}$ corresponds to a
singularity on the real axis.  With this transformation, singularities
on the real axis approach to the origin and those near the imaginary
axis go away.  Since the DLPA can estimate a position of the nearest
singularity most accurately, errors of the DLPA can become smaller by
this transformation.


In order to find an optimal $a$, we calculate $T_{\rm N}$ as a function
 of $a$ for a couple of choices of the DLPA, and we adopt $a$ at which
 difference among different DLPAs is the smallest.  Figure
 \ref{fig:valtran} shows $T_{\rm N}$ as a function of $a$ for three
 different choices of the DLPA.  Here, $[m/n]$ denotes the DLPA with a
 polynomial of order $m$ over a polynomial of order $n$.  Since [4/4] is
 from the series one-order lower than the others, we choose $a$ at which
 [5/4] and [4/5] are the closest, namely, $a=1.04$.  Then, $T_{\rm N}/J$
 obtained here is 1.898, which is close to those in the literature,
 $1.892$ by the QMC\cite{Sandvik98} and $1.888$ by the HTSE\cite{Oitmaa}.
In addition, we have obtained a critical exponent simultaneously.
We assume that the critical exponent does not depend on the spin magnitude,
and compare it with that of the {\em classical} antiferromagnetic Heisenberg model,
in which $S(\pi,\pi,\pi)$ is identical to the staggered susceptibility
equivalent to the `uniform susceptibility of the ferromagnetic model'.
Hence we can compare the critical exponent $\gamma$.
The estimation from our calculation above is $\gamma=1.399$,
which is close to
1.396 by a Monte Carlo method\cite{Campostrini02},
1.406 by HTSE\cite{Butera97}, and 1.388 by a field theory\cite{Jasch01},
of the classical O(3) model.
Therefore, we trust this way of analyzing $T_{\rm N}$,
and use it also for the SU($n$) model with $n>2$.
Also for $n>2$,
the variable transformation (\ref{eq:trans}) should work
because the singularity closest to the origin in the original series is
near the imaginary axis.

\begin{figure}
\begin{center}\includegraphics[  width=8cm
]{valtran}
\caption{\label{fig:valtran}
The N\'eel Temperature $T_{\rm N}$ of the SU(2) Heisenberg model
obtained by three different DLPAs as a function of the parameter $a$.
}
\end{center}\end{figure}

\begin{figure}
\begin{center}\includegraphics[  width=8cm
]{ndep}
\caption{\label{fig:ndep}
The N\'eel Temperature $T_{\rm N}$ of the SU($n$) Heisenberg model
as a function of $n$ with $a=1.04$.
Here, [5/4] and [4/5] are plotted together.
}
\end{center}\end{figure}


Originally $n$ is an integer because it is the number of internal
degrees of freedom.  However, after obtaining the series as an analytic
function of $n$, we can regard $n$ as a continuous variable that has a
physical meaning only when it happens to take integer values.  In fact,
as $n$ changes continuously, the properties of the series change also
continuously. Therefore, we gradually increases $n$ from $n=2$ and see
$n$-dependence of $T_{\rm N}$.  Figure \ref{fig:ndep} shows the results.
Here, we fix $a=1.04$, and we plot [5/4] and [4/5] together.  The
optimal $a$ may depend on $n$.  However, since the difference between
the two curves is very small, we regard it as an optimal $a$.  As $n$
increases, $T_{\rm N}$ decreases almost linearly, and at $n\sim2.45$ the
singularity corresponding to $T_{\rm N}$ runs away from the real axis to
a complex value with a finite imaginary part. For larger $n$, we do
not find any singularity on the antiferromagnetic side of the real axis.
Therefore, we conclude that the value of $n$ at which
N\'eel order disappears lies in the range $2<n<3$.


In addition, we have also analyzed $S(\mathbf{q})$ with
$\mathbf{q}\neq(\pi,\pi,\pi)$. However, we have not found any
symptom of ordering with $\mathbf{q}\neq(\pi,\pi,\pi)$ in the
temperature region that we can reach by the present order of the series.





\section{summary}

In summary, we have performed high-temperature series expansions for
the SU($n$) Heisenberg model in three dimensions with arbitrary $n$.
%
First of all,
we have calculated a \textit{next}-nearest-neighbor correlation
function.  At least at large $n$, it changes the sign at low
temperature, which suggests that the correlation should not be like N\'eel
order, but have a longer period. Analysis of the Fourier transform of the
correlation function also supports that the ground state of the large-$n$
SU($n$) Heisenberg model does not have the N\'eel-order correlation.
%
Then, we have turned to an approach from $n=2$.  Since the SU(2)
Heisenberg model has the N\'eel order, it should disappear at a
certain $n$.  We have first found a reliable way to estimate a
transition temperature by analyzing the divergence of the $(\pi,\pi,\pi)$
component of the correlation function.
Next, we generalize $n$ to a continuous
variable and increase $n$ gradually from $n=2$.
We have concluded that the N\'eel
ordering disappears for $n>2$.




\acknowledgments
The author would like to thank Y.~Kuramoto for
directing him to this topic and for stimulating discussions.
%
He also appreciates useful comments on the manuscript from A.~Honecker.
%
This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG).
%
Early stages of this work were supported by the Japan Society for the
Promotion of Science and by the Max-Planck Institute for the Physics of
Complex Systems.
Different stages of this work were supported by the DFG
and the Technical University Braunschweig.
%
Parts of the numerical calculations were performed on
the {\tt cfgauss} at the computing centers of the TU Braunschweig.


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