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\LARGE \bf
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\begin{center}
{\huge \bf
 New algorithm \\ 
of the high-temperature expansion 
%\hspace*{-0.5cm}
 }
\end{center} 

\huge \bf
%\vspace{-0.2cm}
\hspace*{5.7cm}    T. Fujiwara  \hspace{2.4cm} \\
\hspace*{6cm}      H. Arisue \hspace{1cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{2.0cm}
\hspace*{2cm}
{\huge \bf 1. Introduction}

\hspace*{2cm}
{\huge \bf 2. Finite lattice method }

\hspace*{2cm}
{\huge \bf 3. New algorithm of the finite lattice \\
\hspace*{3.3cm}
 method}

\hspace*{2cm}
{\huge \bf 4. Results}

\hspace*{2cm}
{\huge \bf 5. Summary and Discussions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\huge \bf

\vspace*{-3cm}
\hspace*{-1cm}
{\huge \bf 1. Introduction}

\begin{itemize}
%%%%%%%%%
%\item[$\bullet$]
%��͓I��(�W�J)���@
%
%\hspace*{1cm}
%       e.g. �����W�J, �ቷ�W�J, $\epsilon$-�W�J
%
%\hspace*{1cm}
%       ���l�V�~�����[�V�����Ƒ���I�Ȋ֌W
%
%\hspace*{2cm}
%          �萫�I �� \underline{��ʓI}

\item[$\bullet$]
Analytic parameter expansion

\hspace*{0.3cm}
    e.g. -- High-temperature expansion\\
\hspace*{1.95cm}
         -- Low-temperature expansion\\
\hspace*{1.95cm}
         -- Strong coupling expansion\\
\hspace*{1.95cm}
         -- Hopping parameter expansion\\ 
\hspace*{2.0cm}
       $\Rightarrow$ not only qualitative \\ 
\hspace*{3.0cm}
       but also quantitative results,\\
\hspace*{3.0cm}
       if we have long enough series.

\item[$\bullet$]
High-temperature expansion of Spin \\
models

%���������܂Ōv�Z����̂ɕK�v�Ȍv�Z�ʂ�,

\hspace*{0.3cm}
     ��In two dimensions

\hspace*{2cm}
          --  Diagrammatic method :

\hspace*{4cm}
                $ T_{calc}$ : Total amount of \\
\hspace*{6.5cm}
                the calculation

\hspace*{5.3cm}
                $ \sim \exp{(a N)}$

\hspace*{2cm}
          --  \fbox{Finite lattice method} :

\hspace*{4cm}
                $ T_{calc} \sim \exp{(b N)}$

\vspace{-0.5cm}
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%\hspace*{1cm}
%            Finite lattice method   ����  Diagrammatic method   

%\vspace{0.5cm}
%\hspace*{1cm}
%      e.g. 2����$Q$-state Potts�͌^�� large $Q$ �W�J

%\hspace*{2cm}
%      �E10�� : Bhattacharya {\em et al.} (1997) 

%\hspace*{4.5cm}
%         �E�E�E�E Diagrammatic method 

%\hspace*{2cm}
%       �E23�� : K. Tabata and H.A. (1999)\\
%\hspace*{4.5cm}
%        �E�E�E�E�E Finite lattice method

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

%\hspace*{0.3cm}
     ��In three dimensions

\hspace*{1cm}
          --  Diagrammatic method : 

\hspace*{3cm}
            $T_{calc} \sim \exp{(a N)}$

\hspace*{1cm}
         --   Finite lattice method :

\hspace{3cm}  $T_{calc} \sim \exp{(b N^2)}$

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\vspace{1cm}
\hspace*{7cm}$\Downarrow$

\vspace{1cm}
\hspace*{1cm}
            $\diamondsuit$ \fbox{New algorithm} of the finite lattice\\
\hspace*{2cm}
             method for the Ising model

\vspace{0.7cm}
\hspace*{4cm}
             $T_{calc} \sim  \exp{(c N\log N)}$

\vspace{0.7cm}
\hspace*{7cm}
             


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\newpage
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\newpage
\hspace*{-2cm}
Orders of the high-temperature expansion \\
\hspace*{-1cm}
for the Ising model in three dimensions

\begin{table}[h]
\hspace*{-1cm}
%\caption{
%         }
%\begin{center}
\begin{tabular}{c|c|c|c}
\hline
         &     &  \multicolumn{2}{|c}{Finite lattice method}   \\
\cline{3-4}
         &  \raisebox{0mm}{Diagrammatic} &  old &  new  \\
         &   method                     & algorithm & algorithm \\
\hline
 free energy  &     $N=$ 18      &     $N=$ 26   &    $N=$ 46  \\
\hline
 susceptibility  &     $N=$ 25       &     $N=$ 13   &    $N=$ 32  \\
\hline
\end{tabular}
%\end{center}
\end{table}

%\vspace{1.5cm}
%\hspace*{1cm}
%      e.g. 3�����C�W���O�͌^�̍����W�J

\hspace*{2cm}
      \item free energy

\vspace{-0.3cm}
%\hspace*{3cm}
%       24�� : Bhanot, Creutz, Gl\"{a}ssner, Schilling \\
\hspace*{3cm}
       $N=$ 26 : Guttmann and Enting \\
\hspace*{4.5cm}
        (1994) $\cdots$ Finite lattice method

\hspace*{2cm}
      \item magnetic susceptibility

\vspace{-0.3cm}
\hspace*{3cm}
          $N=$ 25 : Butera and Comi \\ 
\hspace*{4.5cm}
        (2002) $\cdots$  Diagrammatic method 

%\newpage
%      3�����ŁuFinite lattice method ����  Diagrammatic method�v�łȂ����R


%\hspace*{1cm}
%            ��3�����C�W���O�͌^�̍����W�J�ɓK�p\\
%\hspace*{1.7cm}
%            �E���R�G�l���M�[(��M)

%\hspace*{3cm}
%                   46��

%\hspace*{1.7cm}
%            �E�ю���

%\hspace*{3cm}
%                   32��
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%>>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\vspace*{-1cm}
\hspace{-1.5cm}
$\diamondsuit$ In the finite lattice method (with the old algorithm)

\item 
%\vspace{1cm}
maximum  size of the lattice whose partition function is calculable:

\hspace{2cm}
$\l_x \times l_y\times l_z = 4 \times 4 \times 5 $ 

%%%%%%%%%%%%%%%%%%%%%
\vspace*{-0.5cm}
\hspace{-0.5cm}
\item 
free enenrgy

\hspace{0.5cm}
$ N = 2 (l_x+l_y+l_z) = 26$

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\vspace{-3.5cm}
\hspace*{-0.5cm}
\item
magnetic susceptibility

%\hspace*{0.5cm}
%maximum  size of the lattice calculable:

%\vspace{1cm}
%\hspace*{2cm}
%$4 \times 4 \times 5 $ 

\hspace*{0.5cm}
$ N = l_x+l_y+l_z = 13$

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%>>
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\newpage
\vspace*{-1.5cm}
\hspace*{-2cm}
%\section{Finite lattice method}
{\huge \bf 2. Finite lattice method}

\item 
%     Series expansion by finite lattice method

         \begin{description}
         \item
             Low- and high-temperature expansion in \\
\hspace*{-0.8cm}
             spin systems

             \begin{itemize}
                \item
                 Enting (1978) 
                \item
                 Creutz (1991)   
             \end{itemize}
        \end{description}
\item 
         \begin{description}
         \item
             Strong coupling expansion in lattice  \\
\hspace*{-0.8cm}
            gauge theory
             \begin{itemize}
                \item
                 Fujiwara and H.A. (1984)
             \end{itemize}
        \end{description}
                   \hspace{5cm}��

\hspace*{-1cm} $<$ Ising model in two dimensions $>$

%\item 
%     Finite lattice method�ɂ�鋉���W�J
%
%         \begin{description}
%         \item
%             �ቷ�W�J�E�����W�J(�X�s�����v�n)

%             \begin{itemize}
%                \item
%                 Enting (1978) 
%                \item
%                 Creutz (1991)   
%             \end{itemize}
%         \item
%             �������W�J(�i�q�Q�[�W���_)
%             \begin{itemize}
%                \item
%                 Fujiwara and H.A. (1984)
%             \end{itemize}
%        \end{description}
%                   \hspace{5cm}��

\vspace{0.5cm}
Partition function
$$
 Z(\beta)=\sum_{\{ s_i \}} 
  \exp{\left( \beta\; \sum_{i,j} s_i s_j 
  %+ \mu\; \sum_i s_i 
  \right)} 
$$

%Susceptibility
%$$
%\chi=\frac{\partial^2}{\partial \mu^2} 
%   \log{\left[ Z(\beta,\mu)\right]}
%$$

\vspace{0.8cm}
Boltzmann factor
$$
 \exp{\left( \beta\; s_i s_j \right)} 
  = \cosh{(\beta)}\; (1\; +\; t\; s_i s_j)
$$
$$
      t = \tanh{(\beta)} 
$$
$$\left( \beta \equiv \frac{J}{k_{B}T} \right)$$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\item[ $\diamondsuit$ ]
Finite lattice method\\
\hspace*{2cm}
(= non-diagramatic method)

\hspace*{1cm}
\item
        Calculate the partition function  
        $Z(l_x\times l_y)$ \\
%\hspace*{1.5cm}
        for the finite size lattices with $2(l_x+l_y) \le N$\\
        to order $t^N$

\hspace*{1cm}
\item
        Calculate $\phi(l_x \times l_y)$ %to order $t^N$ 
        recursively as


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\vspace{-0.5cm}
\hspace*{-1cm}
$
    \qquad\quad \phi(1 \times 1) = \log{Z(1 \times 1)} 
$

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\vspace{-0.5cm}
\hspace*{-1.5cm}
$
\qquad\quad \phi(2 \times 1) =  \log{Z(2 \times 1)} - 2 \phi(1 \times 1) 
$


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\vspace{-1.5cm}
\begin{eqnarray}
\quad \phi(2 \times 2) &=& \log{Z(2 \times 2)} \nonumber\\
&& -2\phi(2 \times 1)-2\phi(1 \times 2)-4\phi(1 \times 1) \nonumber
\end{eqnarray}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\vspace{0.5cm}
For general $l_x \times l_y$ lattice, 
{\LARGE \bf
\begin{eqnarray} 
\phi(l_x \times l_y) &=& {\log{Z(l_x \times l_y)}} \nonumber\\
    && - \sum_{\scriptstyle l_x^{\prime} \le l_x,l_y^{\prime}\le l_y
              \atop\scriptstyle 
              (l_x^{\prime},l_y^{\prime}) \ne (l_x,l_y)}
           (l_x-l_x^{\prime}+1)(l_y-l_y^{\prime}+1)
            \phi(l_x^{\prime} \times l_y^{\prime}) \nonumber
\end{eqnarray} 
}

Then, the free energy density in the infinite volume limit:
$$
f=\sum_{2(l_x+l_y) \le N} \phi(l_x,l_y)
$$

\vspace{1cm}
\hspace*{7cm}
          $\Uparrow$     


The Taylor expansion of $\phi(l_x \times l_y)$ w.r.t. $t$ 
gives automatically the contribution 
to the free energy density in the infinite \\ 
volume limit 
from all the connected diagrams (clusters of polymers) 
that just fit the $l_x \times l_y$ lattice.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1cm}
%In the infinite volume limit
%$$
%    \log{Z(V)} = V \sum_{l_x \times l_y} \phi(l_x  \times l_y) 
%$$
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\vspace{1cm}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\item
        Partition function $Z(l_x \times l_y)$ is
        calculated by the transfer matrix formalism

%\hspace*{1.5cm}
        (in bond-by-bond construction)

\hspace*{3cm}
        $Z=V_{0i}V_{ij}V_{ik}\cdots V_{l0}$\,.

%\hspace*{1.5cm}
       \underline{All of the configurations} with the \\
order $n\le N$
are taken into account 
\underline{in the}\\
\underline{finite lattice method}.\\
       
\vspace{0.5cm}
\item
Maximum size of the lattice to be considered:
\begin{eqnarray}
l_x &\sim & l_y \sim \frac{N}{4} \quad
       \mbox{in two dimensions}\nonumber\\
l_x &\sim &l_y \sim l_z \sim \frac{N}{6} \quad
       \mbox{in three dimensions}\nonumber
\end{eqnarray}

\vspace{0.5cm}
\hspace*{1cm}
\item
        Total amount of the calculation\\
%        \hspace*{2cm}
%        $\sim$ the amount of calculation \\
%        \hspace*{3cm}
%        for the transfer matrix element\\
        \hspace*{1cm}
        $\propto$ the number of states $\{i\}$

        \hspace*{1cm}
         $ \left\{
           \begin{array}{@{\,}l}
              \propto 2^{l_x} \sim \exp{(cN)} \ \ \ \mbox{in two dimensions}\\
               \propto 2^{l_x\times l_y} \sim \exp{(c^{'}N^2)} \ \ \ 
                   \mbox{in three dimensions}
           \end{array} \right.   $ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%\item[��]
%Finite lattice method
%
% \begin{enumerate}
%   \item
%   ���z�֐� ${Z(1 \times 1)}$, ${Z(2 \times 1)}$, ${Z(2 \times 2)}$,
%   �E�E�E �̌v�Z
%
%   \item
%   �ϊ�\\
%   $$\phi(l_x \times l_y) = {\log{Z(l_x \times l_y)}}
%        - \sum_{\scriptstyle l_x^{\prime} \le l_x,l_y^{\prime}\le l_y
%              \atop\scriptstyle 
%              (l_x^{\prime},l_y^{\prime}) \ne (l_x,l_y)}
%           (l_x-l_x^{\prime}+1)(l_y-l_y^{\prime}+1)
%            \phi(l_x^{\prime} \times l_y^{\prime})  $$  
%
%
%   \item
%   �����n�̎��R�G�l���M�[���x\\
%   $$ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
%       \lim_{V \rightarrow \infty} \frac{1}{V}\log{Z(V)} 
%         = \sum_{l_x \times l_y} \phi(l_x  \times l_y) $$
% \end{enumerate}
%
%    \vspace{1cm}
%    \begin{description}
%        \item[$\bullet$]
%        �B��K�v�ȃ_�C�A�O�����̒m�� :
%
%        $\phi(l_x  \times l_y)$ �Ɋ�^����Œ᎟��
%        �_�C�A�O�����͉���.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    \vspace{0.5cm}
%         $$\qquad\phi(l_x  \times l_y) = O (\ t^{2(l_x +l_y)}\ ) $$
%    \end{description}
%
% \hspace{1cm}
%   ��  $\displaystyle l_x +l_y \le N/2$ �𖞂����i�q�T�C�Y�ɂ��Ă���\\
% \hspace{1cm}
%      ���z�֐����v�Z���Ă����΂悢.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item
%���z�֐��̌v�Z
%
%      bond-by-bond construction
%           
%    \hspace{1cm}
%         �E multiply the factor $\displaystyle 1\ +\ t\; s_{i}s_{j}$
%
%    \hspace{1cm}
%         �E integrate {\huge $\displaystyle\sum_{s_i=\{-1,+1\}}$}
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%
%    \vspace{1.5cm}
%    \hspace{3cm}
%         �r���̒f�ʂ̏�Ԑ� = $2^{\; l_x}$  $\sim 2^{N/4} $
%
%    \hspace{3cm}
%         �v�Z�� = $ l_x\; l_y\; 2^{\; l_x}$ $\sim 2^{N/4} $
%         $$ ( l_x + l_y \le N/2 ) $$
%
%\vspace{1.5cm}
%\hspace{1cm}
%3�����̏ꍇ
%
%    \hspace{3cm}
%         �r���̒f�ʂ̏�Ԑ� = $2^{\; (l_x +1)\;(l_y+1)-1}$ 
%                $\sim 2^{N^2/9} $
%
%    \hspace{3cm}
%         �v�Z�� = $ l_x\; l_y\; l_z\; 2^{\; (l_x +1)\;(l_y+1)-1}$ 
%        $\sim 2^{N^2/9} $
%         $$( l_x + l_y + l_z\le N/3 ) $$
%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hspace*{-1.5cm}
{\huge \bf 3. New algorithm of the finite lattice method}

\item
In order to obtain the series to a given order, however, 
it is enough to consider only a restricted number of 
bond configurations for $Z(l_x \times l_y)$.

i.e. Most of the bond configurations do not contribute 
to $\phi(l_x \times l_y)$ and can be neglected systematically.

\vspace{1cm}
\item
Consider the anisotropic model\ \ ($t_x \ne t_y$) in two dimensions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To obtain the series to order \\
\hspace*{3cm} $N=N_x +N_y=20$,\\
the largest size of the lattice is \\
\hspace*{3cm} $l_x\times l_y=5\times 5$ \\
and\\
\hspace*{3cm}  $N_x=2l_x=10$,\\
\hspace*{3cm}  $N_y=2l_y=10$.

%$\phi(5\times 5)$ 


\newpage
\item
 Configurations for  $Z(l_x\times l_y)$ 
that contribute to the lowest order term of  
$\phi(5\times 5)$ in $t_y^{\ N_y}$\\
($N_y=2 l_y=10$)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{1cm}
\hspace*{-2cm}
$Z(l_x\times l_y)$

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\put( 90,-20){\mbox{$\sum_i \max(n_{i},2)$ =10}}
%%%%%%%%%%%%%%%%%%%%
\end{picture}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4cm}
\hspace*{-1cm}
$\phi(l_x\times l_y)$

%%%%%%%%%%%%%%
\begin{figure}[h]
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\begin{picture}(110,50)
%%%%%%%%%%%%%%
\put(0,0){\begin{picture}(50,50)(  0,0)
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%%%%%%%%%%%%%%%%%%%%
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\put(100,-10){\mbox{$N_y$ =10}}
%%%%%%%%%%%%%%%%%%%%
\end{picture}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%>>
\begin{figure}[h]
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%%%%%%%%%%%%%%%%
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\put(0,0){\vector(1,0){50}}
\put(0,0){\vector(0,1){20}}
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\put(0,30){\mbox{y}}
\end{picture}
%%%%%%%%%%%%%%%<<
\end{figure}
%%%%%%%%%%%%%%%<<

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
 Configurations for  $Z(l_x\times l_y)$  
that do \underline{not} contribute to the lowest order term 
of $\phi(l_x\times l_y)$ in $t_y$ 
($N_y=2 l_y=10$)
            
\vspace{1cm}
\hspace*{1cm}
$Z(l_x\times l_y)$
%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\setlength{\unitlength}{1mm}
\begin{picture}(110,50)
%%%%%%%%%%%%%%
\put(20,0){\begin{picture}(50,60)(0,0)
%
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\end{picture}
}

%%%%%%%%%%%%%%%%%%%%
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%
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%
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\put( 40,-10){\mbox{$N_y=10$}}
\put( 25,-20){\mbox{$\sum_i \max(n_{i},2)$ =12}}

%%%%%%%%%%%%%%%%%%%%%
\end{picture}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{4cm}
\hspace*{1cm}
$\phi(l_x\times l_y)$
            
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
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\begin{picture}(110,50)
%%%%%%%%%%%%%%
\put(80,70){\mbox{$\Downarrow$}}

\put(60,0){\begin{picture}(50,50)(0,0)

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\multiput(40,0)(0,2){25}{\line(0,1){1}}
\multiput(50,0)(0,2){25}{\line(0,1){1}}
%
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\end{picture}

}

\put( 70,-10){\mbox{$N_y=12$}}
%%%%%%%%%%%%%%%%%%%%%
\end{picture}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item
\vspace*{-2cm}
%%%%%%%%%%%%%%%<<
%\vspace{2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The second largest lattice size is \\
\hspace*{3cm} $l_x\times l_y=4\times 5$ \\
and\\
\hspace*{3cm}  $N_x=2l_x=8$, \\
\hspace*{3cm} $N_y=2l_y+2=12$.

\vspace{1cm}
$Z(l_x\times l_y)$
%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\setlength{\unitlength}{1mm}
\begin{picture}(110,50)
%%%%%%%%%%%%%%
\put(20,0){\begin{picture}(50,60)(0,0)
%
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\put(42,12){\mbox{2}}
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\multiput(30,0)(0,2){25}{\line(0,1){1}}
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%
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\put( 10,-20){\mbox{$\sum_i \max(n_{i},2)$ =12}}

%%%%%%%%%%%%%%%%%%%%
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%
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\put( 80,-20){\mbox{$\sum_i \max(n_{i},2)$ =12}}
%%%%%%%%%%%%%%%%%%%%
\put(130,0){\begin{picture}(50,50)(0,0)
\put(10,34){\mbox{should be}}
\put(10,24){\mbox{included}}
\end{picture}
}
%%%%%%%%%%%%%%%%%%%%%
\end{picture}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\vspace{5cm}

%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\setlength{\unitlength}{1mm}
\begin{picture}(110,50)
%%%%%%%%%%%%%%
\put(20,0){\begin{picture}(50,60)(0,0)
%
\put(42, 2){\mbox{0}}
\put(42,12){\mbox{4}}
\put(42,22){\mbox{4}}
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%
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\put( 25,-10){\mbox{$N_y$ =10}}
\put( 10,-20){\mbox{$\sum_i \max(n_{i},2)$ =14}}
%%%%%%%%%%%%%%%%%%%%
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%
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\put(42,12){\mbox{4}}
\put(42,22){\mbox{4}}
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\multiput(30,0)(0,2){25}{\line(0,1){1}}
\multiput(40,0)(0,2){25}{\line(0,1){1}}
%
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\end{picture}
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\put( 90,-10){\mbox{$N_y$ =10}}
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%%%%%%%%%%%%%%%%%%%%
\put(130,0){\begin{picture}(50,50)(0,0)
\put(10,34){\mbox{can be}}
\put(10,24){\mbox{neglected}}
\end{picture}
}
%%%%%%%%%%%%%%%%%%%%%
\end{picture}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\vspace*{-2cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%>>
\begin{figure}[h]
\setlength{\unitlength}{1mm}
%%%%%%%%%%%%%%%%<<
\begin{picture}(170,78)(0,0)
\put(0,77){\line(1,0){175}}
\put(0, 0){\line(1,0){175}}
\put(  0,0){\line(0,1){77}}
\put(175,0){\line(0,1){77}}
\put(10, 62){\huge \sf \bf 
To obtain $\phi(l_x\times l_y)$ in the order of $N_y$ in $t_y$\,,} 
\put( 6,48){\huge \sf \bf 
only the configurations for $Z(l_x\times l_y) $ that 
}
\put( 6,35){\huge \sf \bf 
have the order $n_{i}$ of the $i$-th layer that satisfy 
}
\put( 6,22){\huge \sf \bf 
\hspace*{2cm}  $\displaystyle 
\sum_{i=1}^{l_y} \max(n_{i},2) \le N_y$
}
\put( 6, 8){\huge \sf \bf 
    should be taken into account.
}
\end{picture}
%%%%%%%%%%%%%%%<<
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{0.5cm}
\hspace*{-2cm}
$\diamondsuit$ In three dimensions

\vspace{-1cm}
\hspace*{1cm}
\item
Largest size of the lattice to be considered:

\hspace*{2cm}
 {\huge $l_x\sim l_y\sim l_z\sim N/6$}

\hspace*{4cm}
 { with \huge $2(l_x+l_y+l_z)=N$}

%\vspace*{-0.5cm}
\item
Partition function is calculated by the \\ transfer matrix formalism
taking into account only the bond configurations with
\hspace*{4cm}
   {\Huge  $n_{zi}\le 2$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
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\vspace*{-1cm}
\begin{figure}[htb!]
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\vspace{-1cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
      Z=V_{0,2}\;\;V_{2,2}\;\;\cdots 
          \;\;V_{2,2}\;\;V_{2,0}
$$
%\vspace{1cm}
%\hspace{1cm}
    --  Transfer matrix $V_{2,2}$ is like 
%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
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\put(35,0){\begin{picture}(50,30)(0,0)

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\end{picture}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%

\vspace{1.5cm}
This transfer matrix itself is the 'partition function' in two dimensions
and can be 
calculated by another transfer matrix
formalism.


\vspace{-0.5cm}
\hspace*{1cm}
\item
Amount of the calculation for the transfer matrices

\hspace*{1cm}
{\huge $ \propto  
   C(l_x\;l_y,\;4)\; 2^{\;l_x} 
    \propto N^4 2^{N/6}
    \sim  \exp{(cN+c'\log{N})}$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%>>
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{-2cm}
\hspace*{-1cm}
{\Huge\bf 4. Results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{-1cm}
\hspace{2cm}
\item free energy

\vspace{-1cm}
          {\huge 
          $$ f = 3 \cosh{(\beta)} + \sum_n a_n t^n $$
          $$ t = \tanh{(\beta)} $$ }

\vspace{-0.5cm}
\LARGE{
\vspace{-1cm}
\begin{table}[h]
%\caption{
%         }
\begin{center}
\begin{tabular}{r|l}
\hline
    $n$  & \multicolumn{1}{c}{$a_n$}   \\
\hline
$  2$ & $ 0                             $  \\ 
$  4$ & $ 3                             $  \\ 
$  6$ & $ 22                            $  \\ 
$  8$ & $ 375/2                         $  \\ 
$ 10$ & $ 1980                          $  \\ 
$ 12$ & $ 24044                         $  \\ 
$ 14$ & $ 319170                        $  \\ 
$ 16$ & $ 18059031/4                    $  \\ 
$ 18$ & $ 201010408/3                   $  \\ 
$ 20$ & $ 5162283633/5                  $  \\ 
$ 22$ & $ 16397040750                  $  \\ 
$ 24$ & $ 266958797382                 $  \\ 
$ 26$ & $ 4437596650548                $  \\ 
\hline
$ 28$ & $ 525549581866326/7            $  \\ 
$ 30$ & $ 6448284363491202/5           $  \\ 
$ 32$ & $ 179577198475709847/8         $  \\ 
$ 34$ & $ 395251648062268272           $  \\ 
$ 36$ & $ 21093662188820520521/3       $  \\ 
$ 38$ & $ 126225408651399082182        $  \\ 
$ 40$ & $ 4569217533196761997785/2     $  \\ 
$ 42$ & $ 291591287110968623857940/7   $  \\ 
$ 44$ & $ 8410722262379235048686604/11 $  \\ 
$ 46$ & $ 14120314204713719766888210   $  \\ 
\hline
\end{tabular}
\end{center}
\end{table}
}



\newpage
\vspace*{-1.5cm}
\huge
\item 
Critical exponent {\huge $\alpha$} of the specific heat 

{\huge 
\begin{eqnarray} 
C &=& \beta^2 \frac{d^2 f}{d\beta^2} \nonumber\\
  &\sim& (\beta_c-\beta)^{-\alpha} 
     \left[ 1 + \mu(\beta_c-\beta)^{\Delta}
      + \cdots \ \ \right] 
    \nonumber
\end{eqnarray} 

\hspace*{4cm}
$\displaystyle
\beta_c=  \left\{
\begin{array}{l}
          0.2216546(\ 1) \\
          0.2216595(15) \\
\end{array} \right.
$

\hspace*{4cm}
$\displaystyle \Delta \sim 0.5$ \\
\hspace*{4.5cm}
   (correction-to-scaling exponent)

\begin{eqnarray}
 \alpha_n &\equiv& [\beta_c^2 \frac{a_{2n}}{a_{2n-2}} -1] n +1 
                     \nonumber\\
          &\sim  & \alpha + \frac{b}{n^{\Delta}} + \frac{c}{n} 
             + \cdots \ \ 
                     \nonumber
\end{eqnarray} 
}


\begin{table}[h]
\begin{center}
\begin{tabular}{c|c|c}
\hline
                          &               & {\huge $\alpha$} \\
\hline
                          &  $\beta_c=0.2216546(\ 1)$  & {\huge $0.1035 \pm 0.0010$}  \\ 
\cline{2-3}
 \raisebox{3mm}{H-T expansion} & $\beta_c=0.2216595(15)$   & {\huge $0.1082 \pm 0.0010$}  \\ 
\hline
 \multicolumn{2}{c|}{Monte Carlo}  & {\huge $0.125 \pm 0.025$}  \\ 
\hline
 \multicolumn{2}{c|}{$\epsilon$-expansion} & {\huge $0.1088 \pm 0.0039$}  \\ 
\hline
 \multicolumn{2}{c|}{hyper scaling relation}  & {\huge $0.1096 \pm 0.0005$}  \\ 
\hline
\end{tabular}
\end{center}
\end{table}

$\alpha=2-d\nu$ with $d=3$ and \\
$\nu$: critical exponent for the correlation length 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%   
\vspace*{-2cm}
\hspace{1.6cm}
Magnetic susceptibility
\vspace*{-0.5cm}
\Large \bf
\begin{table}[h]
%\caption{
%         }
\begin{center}
\begin{tabular}{r|r}
\hline
    $n$  & \multicolumn{1}{c}{$a_n$}   \\
\hline
$    0  $  &  $ \qquad                         1$   \\
$    1  $  &  $ \qquad                         6$   \\
$    2  $  &  $ \qquad                        30$   \\
$    3  $  &  $ \qquad                       150$   \\
$    4  $  &  $ \qquad                       726$   \\
$    5  $  &  $ \qquad                      3510$   \\
$    6  $  &  $ \qquad                     16710$   \\
$    7  $  &  $ \qquad                     79494$   \\
$    8  $  &  $ \qquad                    375174$   \\
$    9  $  &  $ \qquad                   1769686$   \\
$   10  $  &  $ \qquad                   8306862$   \\
$   11  $  &  $ \qquad                  38975286$   \\
$   12  $  &  $ \qquad                 182265822$   \\
$   13  $  &  $ \qquad                 852063558$   \\
\hline                   
$   14  $  &  $ \qquad                3973784886$   \\
$   15  $  &  $ \qquad               18527532310$   \\
$   16  $  &  $ \qquad               86228667894$   \\
$   17  $  &  $ \qquad              401225368086$   \\
$   18  $  &  $ \qquad             1864308847838$   \\
$   19  $  &  $ \qquad             8660961643254$   \\
$   20  $  &  $ \qquad            40190947325670$   \\
$   21  $  &  $ \qquad           186475398518726$   \\
$   22  $  &  $ \qquad           864404776466406$   \\
$   23  $  &  $ \qquad          4006394107568934$   \\
$   24  $  &  $ \qquad         18554916271112254$   \\
$   25  $  &  $ \qquad         85923704942057238$   \\
\hline                   
$   26  $  &  $ \qquad        397637244058624494$   \\
$   27  $  &  $ \qquad       1839992653230056950$   \\
$   28  $  &  $ \qquad       8509528288325589438$   \\
$   29  $  &  $ \qquad      39350934581190850230$   \\
$   30  $  &  $ \qquad     181885145332015353030$   \\
$   31  $  &  $ \qquad     840628109226856546326$   \\
$   32  $  &  $ \qquad    3883554493872938687622$   \\
\hline
\end{tabular}
\end{center}
\end{table}



\newpage

\LARGE \bf
\hspace*{-1.5cm}
$\diamondsuit$ Analysis of the series (ratio method�j

{\huge 
\begin{eqnarray} 
\chi &\sim& (\beta_c-\beta)^{-\gamma} 
     \left[ 1 + \mu(\beta_c-\beta)^{\Delta}
      + \cdots \ \ \right] 
    \nonumber
\end{eqnarray} 

%\hspace*{4cm}
%$\displaystyle \beta_c=0.2216546(1)$ \\
\hspace*{4cm}
$\displaystyle \Delta \sim 0.5$ \\
\hspace*{4.5cm}
(correction-to-scaling exponent)

{\huge \bf
\item Critical point {\huge $\beta_c$}}

{\huge 
\begin{eqnarray}
 (\beta_c)_n &\equiv& \left(\frac{a_{n-2}a_{n-3}}{a_{n}a_{n-1}}\right) 
         \exp{\left[\frac{(s_{n} + s_{n-2})}{2s_{n}(s_{n} -s_{n-2})}\right] }
                     \nonumber\\
          &\sim  & \beta_c + \frac{b}{n^{1+\Delta}} 
              + \cdots \ \ 
                     \nonumber
\end{eqnarray} 
}

{\huge 
\begin{eqnarray}
 s_n = \left( \ln{(\frac{a_{n-2}^2}{a_{n}a_{n-4}}})^{-1}
            + \ln{(\frac{a_{n-3}^2}{a_{n-1}a_{n-5}}})^{-1}  \right)/2
                     \nonumber
\end{eqnarray} 
}

\huge
\begin{table}[h]
%\caption{
%         }
\begin{center}
\begin{tabular}{c|c|l}
\hline
 H-T expansion                   &   $N=$ 32     &  $0.2216550(\ 5)$  \\
\hline
 \multicolumn{2}{c|}{}                            &  $0.2216546(\ 1)$   \\
\cline{3-3}
 \multicolumn{2}{c|}{\raisebox{3mm}{Monte Carlo}}  &  $0.2216595(15)$   \\
\hline
\end{tabular}
\end{center}
\end{table}


\newpage
\item  {\huge Critical exponent $\gamma$}

{\huge 
\begin{eqnarray}
 \gamma_n &\equiv& 1 + \frac{2(s_{n} + s_{n-2})}{(s_{n} -s_{n-2})^2} 
                     \nonumber\\
          &\sim  & \gamma + \frac{b}{n^{\Delta}} 
              + \cdots \ \ 
                     \nonumber
\end{eqnarray} 
}

\begin{table}[h]
%\caption{
%         }
\begin{center}
\begin{tabular}{c|c|l}
\hline
                             &   $N=$ 25     &  $1.2378(10)$  \\
\cline{2-3}
 \raisebox{3mm}{H-T expansion}    &   $N=$ 32    &  $1.2370(\ 2)$  \\
\hline
 \multicolumn{2}{c|}{perturbative RG}       &  $1.2396(13)$   \\
\hline
 \multicolumn{2}{c|}{$\epsilon$-expansion}       &  $1.2380(50)$   \\
\hline
 \multicolumn{2}{c|}{Monte Carlo(FSS)}     &  $1.2353(25)$   \\
\hline
 \multicolumn{2}{c|}{H-T expansion } &                  \\
 \multicolumn{2}{c|}{ (generalized Ising model):20th} &  $1.2371(\ 4)$   \\
\hline
\end{tabular}
\end{center}
\end{table}



%\item
%Proposal on CP-PACS :
%
%  �T�U���̌v�Z
%       
%\hspace*{1cm}
%       �ő�i�q�T�C�Y  $9\times 9\times 10$
%
%\hspace*{1cm}
%       Finite lattice method�̐V�����A���S���Y���ł�
%
%\hspace*{2cm}
%         �r���̒f�ʂ̏�Ԑ� = $C(100,2)$ = 5,000
%
%\hspace*{2cm}
%         �]���s�� $V_{2,2}$ �̃T�C�Y = $25 \times 10^{6}$
%
%\hspace*{2cm}
%         �ő�K�v������  �` 45GB
%
%\hspace*{2cm}
%         �v�Z���� �` 400hour $\times$ 1000PU

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\vspace*{-1.5cm}
\hspace*{-1cm}
{\huge \bf 5. Summary and Discussions}
\item
  A new algorithm of the finite lattice method has been presented 
  to generate 
  long series of the high-temperature expansions for the Ising 
  model in three dimensions.\\ 

\vspace{-0.5cm}
\item
  It has been applied to:\\
\hspace*{1cm}
       --  free energy \ \ \  $N=26 \rightarrow N=46$ \\
\hspace*{2.5cm}
           ($\rightarrow  N=50$ calculating on CP-PACS) \\
\hspace*{1cm}
       --magnetic susceptibility \   $N=25 \rightarrow N=32 \\
\hspace*{5cm}
            (\rightarrow  N=40)$\\ 

\vspace{-0.5cm}
\item
  It can also be applied to other quantities: i.e.\\
\hspace*{1cm}
        -- second moment correlation length\\
\hspace*{1cm}
        -- true correlation length\\

\item
  It can also be applied to the models with a continuous symmetry: i.e.\\
\hspace*{1cm}
        -- X-Y model \\
\hspace*{2.5cm}
        (Space Shuttle ${}^4$He experiment \\
\hspace*{4.5cm}
          $\alpha=-0.01056(38)$ )

\item
  It can also be applied to other parameter expansions: i.e.\\
\hspace*{1.7cm}
        -- low-temperature expansion\\

}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage  
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
