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 \begin{document}
 \title{Magnetic susceptibility of the 2D Ising model\\ on the finite
 lattice }{}%
 \author{A. I. Bugrij\footnote{Bogolyubov Institute for Theoretical Physics,
 Kyiv, Ukraine}, O. O. Lisovy\footnote{Department of Physics, Taras
 Shevchenko University, Kyiv, Ukraine}}
 \date{}
 \maketitle
 \begin{abstract}
 The generalization of the form factor representation of the 2D
 Ising model correlation function to the case of the
 general disposition of correlating spins on a cylinder is given. The magnetic
 susceptibility on the lattice where one of the dimensions $N$ is
 finite is calculated in both the para- and ferromagnetic regions
 of the Ising coupling parameter. The singularity structure in the
 complex temperature plane and the thermodynamic limit
 $N\rightarrow\infty$ are discussed.
 \end{abstract}

 \section{Introduction}
 The Ising model has long been a subject of great interest.
 The computation of the free energy \cite{Onsager} and
 spontaneous magnetization \cite{Yang}, the series \cite{Montrol,Wu} and
 nonlinear differential equations \cite{Barouch,Jimbo} for the correlation
 functions are the most important advances of the modern mathematical
 physics. The partition function of the 2D Ising model in zero
 field was evaluated exactly
 \cite{McCoyWu} not only in the thermodynamic limit but also for the finite
 lattice with different boundary conditions. The simplicity  of
 the corresponding expressions enables to get an idea
 about the mechanism of the appearance of critical singularities
 in thermodynamical quantities from both mathematical and physical
 points of view.

 Analytical expressions for the thermodynamical quantities, which
 contain the dependence on the size of the lattice, have numerous
 applications. For example, in a computer simulation of
 thermodynamical systems or field models one often needs such
 expressions to estimate the number of the degrees of freedom for
 which discrete numerical model is adequate to initial
 continuous and infinite system. It is worth mentioning that
 modern experiments and technologies often deal with finite-size
 systems. The theoretical analysis of these problems experiences
 the lack of exactly solvable examples.

 In this paper we present exact expressions for the 2-point
 correlation function and the susceptibility of the 2D Ising model
 on the lattice with one finite ($N=\const$) and other infinite
 ($M\to\infty$)
 dimension. These expressions are very similar to well-known form factor
 expansions \cite{Palmev}, \cite{yamada1}.
 We investigate the singularity structure of the susceptibility for the
 finite $N$ and discuss the thermodynamic limit $N\to\infty$.

\section{Correlation function $\langle\sigma(0,0)\sigma(x,0)\rangle$}
 The Ising model on the $M\times N$ square lattice (Fig. 1)
\begin{figure}[h] \begin{center}
 \includegraphics[height=80mm,keepaspectratio=true]
 {Fig1-mse.eps}
 \caption{The numeration of the sites of the lattice and the variants of
 the
 disposition of correlating spins: a) along the cylinder axis,
 b) arbitrary disposition of correlating spins on the lattice.}
 \end{center} \end{figure}
 is defined by the hamiltonian $H[\sigma]$ $$
H[\sigma]=-J\sum_{\rv}\sigma(\rv) (\nabla_x+\nabla_y)\sigma(\rv),
$$
 where two-dimensional vector
$\rv=(x,y)$ labels sites of the lattice: $x=1,2,\ldots,M$,
$y=1,2,\ldots,N$; the Ising spin $\sigma(\rv)$ in each site takes
on the values $\pm1$; $J>0$ is the coupling constant. Shift
operators $\nabla_x$, $\nabla_y$ act as follows $$
\nabla_x\sigma(x,y)=\sigma(x+1,y),\quad
\nabla_y\sigma(x,y)=\sigma(x,y+1). $$ The partition function and
2-point correlation function at the temperature $\beta^{-1}$ are
defined by
\begin{equation}
\label{2.3} Z=\sum_{[\sigma]}\e^{-\beta H[\sigma]},
\end{equation}
\begin{equation}
 \label{2.4}
\langle\sigma(\rv_1)\sigma(\rv_2)\rangle=Z^{-1}\sum_{[\sigma]}\e^{-\beta
H[\sigma]}\sigma(\rv_1)\sigma(\rv_2).
\end{equation}
The summation in these formulae has to be taken over all spin
configurations. We will use the following dimensionless parameters
\begin{equation}
K=\beta J,\quad t=\tanh K, \quad s=\sinh 2K.
\end{equation}

We will consider the lattice with periodic boundary conditions for
both $X$ and $Y$ directions. This gives two equations for
$\nabla_x$, $\nabla_y$ $$ (\nabla_x)^M=1,\quad (\nabla_y)^N=1. $$
For such boundary conditions the partition function (\ref{2.3})
can be expressed in terms of four summands \cite{McCoyWu}
\begin{equation}
\label{2.7} Z=(2\cosh^2 K)^{MN}\cdot
\frac{1}{2}\biggl(Q^{(f,f)}+Q^{(f,b)}+Q^{(b,f)}-Q^{(b,b)}\biggr),
\end{equation}
where each of them is the pfaffian of the operator $\widehat{D}$
(the lattice analogue of the Dirac operator)
\begin{equation}\label{2.8}
Q=\Pf\widehat{D},
\end{equation}
 where
 \begin{equation}\label{2.9}
\widehat{D}= \left(\begin{array}{cccc} 0&1+t\nabla_x&1&1\\
-1-t\nabla_{-x}&0&-1&1\\ -1&1&0&1+t\nabla_{y}\\
-1&-1&-1-t\nabla_{-y}&0
\end{array}\right).
\end{equation}
 The upper indices
$(f,b)$ of the quantities $Q$  in (\ref{2.7}) correspond to
different types (antiperiodic or periodic) of boundary conditions
for the operators $\nabla_x,\nabla_y$ in (\ref{2.9}):
\begin{equation}
(\nabla_x^{(b)})^M=(\nabla_y^{(b)})^N=1,\quad
(\nabla_x^{(f)})^M=(\nabla_y^{(f)})^N=-1.
\end{equation}
When, for example, $M\gg N$ (i.e. torus is reduced into cylinder),
then in the right hand side of (\ref{2.7}) only ``antiperiodic''
term survives:
\begin{equation}\label{2.11}
 Z=(2\cosh^2K)^{MN}Q^{(f,f)}.
 \end{equation}
Since the operator $\widehat{D}$ is translationally invariant, the
pfaffian (\ref{2.8}) can be easily evaluated. After performing the
Fourier transformation, one has the following factorized
representation for the partition function (\ref{2.11})
\begin{equation}\label{2.12}
Z=2^{MN}{\prod_{\qv}}^{(f,f)}(s^2+1-s\cdot\cos q_x-s\cdot\cos
q_y)^{1/2}.
\end{equation}
The upper index  $(f)$  in products (or sums hereinafter) implies
that the components of quasimomentum $q_x$ and $q_y$ in Brillouin
zone run over halfinteger values in the units $2\pi/M$ and
$2\pi/N$ respectively; integer values correspond to the index
$(b)$. For example, $$
{\prod_{q_y}}^{(f)}\F(q_y)=\prod_{l=1}^N\F\biggl(\frac{2\pi}{N}(l+\Pd)
\biggr),\qquad  {\prod_{q_y}}^{(b)}
\F(q_y)=\prod_{l=1}^N\F\biggl(\frac{2\pi}{N}l\biggr). $$
 The product over one of the quasimomentum components in the right
 hand side of (\ref{2.12}) can be evaluated to the explicit form,
 so for the partition function one has
\begin{equation} \label{2.14}
Z=(2s)^{MN/2}{\prod_{q}}^{(f)}\e^{-M\gamma(q)/2}\bigl(1+
\e^{-M\gamma(q)}\bigr),
\end{equation}
 where the function $\gamma(q)$
 is the positive root of the following equation
\begin{equation}
\sinh^2\frac{\gamma(q)}{2}=\sinh^2\frac{\mu}{2}+\sin^2\frac{q}{2},
\label{2.15}
\end{equation}
 where the parameter $\mu$ is the function of $s$
\begin{equation}
\sinh\frac{\mu}{2}=\frac{1}{\sqrt{2}}\bigl(\sqrt{s}-
1/\sqrt{s}\bigr).\label{2.16}
\end{equation}
 For $q\neq0$  $\gamma(q)$ remains positive in the whole range of
 variable $0<s<\infty$, but $\gamma(0)$ changes its sign after
 crossing the critical point $s=1$. Since the product in
(\ref{2.14}) is taken over fermionic spectrum, which does not
contain the value $q=0$, this does not cause any problem here.
 However, the ambiguity in the definition of $\gamma(0)=\pm\mu$
 leads to two different representations for the correlation
 function.

 The sum over spin configurations in the right hand side of
(\ref{2.4}) for the correlation function can also be written in
terms of pfaffians \cite{BuSha}. Corresponding matrices, however,
are not translationally invariant. This fact crucially complicates
the calculations. Nevertheless, the evaluation of the correlation
function can be reduced to the evaluation of the determinant of a
matrix
\begin{equation}
\langle\sigma(0)\sigma(\rv)\rangle=\det A^{(\mathrm
{dim})},\label{2.17}
\end{equation}
 with considerably smaller dimension ${\mathrm {dim}}={\mathrm
 {dim}}(\rv)$, defined by the distance between correlating spins.
 Further work is needed to transform the representation
 (\ref{2.17}) into the representation with analytical dependence
 on distance.

Form factor representation for the correlation function of the
Ising model is the most acceptable from physical point of view.
First it was obtained in \cite{Palmev} for the infinite lattice in
the ferromagnetic region ($K>K_c$, $s>1$). Later it was extended
\cite{yamada1} for the paramagnetic case ($K<K_c$, $s<1$).
 We note that somewhat earlier similar representation for the
 2-point Green function was deduced in \cite{Berg} via
$S$-matrix approach \cite{Zam} for a quantum field model with
factorized $S$-matrix ($S_2=-1$), which is usually associated with
the scaling limit of the Ising model. The discovery of the form
factor representation for the correlation function has led to the
whole trend \cite{Smirnov} in the integrable quantum field theory.

For the finite lattice the problem seems to be more difficult, but
the result \cite{Bugrij} is even simpler in a sense. If
correlating spins are located along one of the axes of the
lattice, the matrix in the right hand side of (\ref{2.17}) is of
Toeplitz form. For example, when correlating spins are located
along the horizontal axis (Fig. 1a), then
\begin{equation}\label{2.18}
\langle\sigma(\rv_1)\sigma(\rv_2)\rangle=\det A^{(|x|)},
 \qquad \rv_2-\rv_1=(x,0),
\end{equation}
$|x|\times|x|$ matrix $ A^{(|x|)}_{k,k'}$ has the elements
\cite{Bugrij}
\begin{eqnarray}&&
A^{(|x|)}_{k,k'}=\frac{1}{MN}{\sum_{\pv}}^{(f,f)}\ \frac
{\e^{ip_x(k-k')}
[2t(1+t^2)-(1-t^2)(\e^{ip_x}+t^2\e^{-ip_x})]}{(1+t^2)^2-2t(1-t^2)
(\cos p_x+\cos p_y)},\label{2.19}\\&&\ \qquad
k,k'=0,\,1,\,\ldots,\,|x|-1.\nonumber
\end{eqnarray}

 As it was shown in \cite{Bugrij} via Wiener-Hopf integral
 equations technique \cite{McCoyWu} adjusted to the finite-sized lattice,
 the determinant (\ref{2.18})
 can be evaluated analytically and for the correlation function
 one has
   \begin{eqnarray}\label{2.20}
\langle\sigma{(\rv_1)}\sigma{(\rv_2)}\rangle&=&(\xi\cdot\xi_T)\e^{-|x|
/\Lambda} \sum_{l=0}^{[N/2]}g_{2l}(x),\quad\hspace{1cm}{\text
{for\ \ }} \gamma(0)=\mu,\\\label{2.21}
\langle\sigma{(\rv_1)}\sigma{(\rv_2)}\rangle&=&
(\xi\cdot\xi_T)\e^{-|x|/\Lambda}
\sum_{l=0}^{[(N-1)/2]}g_{2l+1}(x),\quad{\text {for\ \ }}
\gamma(0)=-\mu,\\\label{2.22} g_n(x)&=&\frac{\e^{-n/\Lambda}}
{n!N^n}{\sum_{[q]}}^{(b)} \biggl(\prod_{i=1}^{n}
\frac{\e^{-|x|\gamma_i-\eta_i}}{\sinh \gamma_i}
\biggr)F_n^2[q],\quad g_0=1,\\\label{2.23} F_n[q]&=&
\prod_{i<j}^{n}\frac{\sin((q_i-q_j)/2)}{\sinh((\gamma_i+\gamma_j)/2)},
\quad F_1=1, \end{eqnarray} where $\gamma_i=\gamma(q_i)$,
$\eta_i=\eta(q_i)$.
   The expressions (\ref{2.20}), (\ref{2.21}) are the finite sums.
   However, upper limits of summation can be set infinite,
   since it follows from (\ref{2.23}) that form factors
   $F_n[q]$ vanishes for $n>N$.
   Note an important detail -- summation over the phase volume in
   (\ref{2.22}) is taken over bosonic spectrum of quasimomenta, in
   contrast with initial fermionic spectrum, which defines the matrix
   (\ref{2.19}). The other quantities in
   (\ref{2.20})--(\ref{2.23}) are given by
\begin{eqnarray}\xi&=&|1-s^{-4}|^{1/4},\label{2.24}\\
\ln\xi_T&=&\frac{N^2}{2\pi^2}\int\limits_{0}^{\pi}\frac{dp\, dq\,
\gamma'(p)\gamma'(q)}{\sinh (N\gamma(p))\sinh
(N\gamma(q))}\ln\biggl|
\frac{\sin((p+q)/2)}{\sin((p-q)/2)}\biggr|, \label{2.25}\\
 \Lambda^{-1}&=&
\frac{1}{\pi}\int\limits_{0}^{\pi}dp\,\ln\coth({N\gamma(p)/2}),
 \label{2.26}\\
\eta(q)&=& \frac{1}{\pi}\int\limits_{0}^{\pi}\frac{dp\:(\cos p-
\e^{-\gamma(q)})}{\cosh\gamma(q)-\cos p}\ln\coth (N\gamma(p)/2).
 \label{2.27}\end{eqnarray}

 ``Cylindrical parameters'' $\xi_T$,
$\Lambda^{-1}$, $\eta(q)$  explicitly depend on the number of
sites $N$ on the base of the cylinder. Their asymptotic behaviour
for $N|\mu|\gg1$ is following
\begin{eqnarray}\label{2.28}
\ln\xi_T &\simeq&\frac{1}{\pi}\e^{-2N|\mu|},\\
\Lambda^{-1}&\simeq& e^{-N|\mu|}
 \sqrt{\frac{2\sinh|\mu|}{\pi N}}\label{2.29}\\
 \eta(q)&\simeq&\frac{4\e^{-N|\mu|}}{(\e^{\gamma(q)}-1)}
\sqrt{\frac{\sinh|\mu|}{2\pi N}}.\label{2.30}\end{eqnarray}
  Outside the critical point cylindrical parameters
 $\Lambda^{-1}$, $\ln \xi_T$ and $\eta(q)$ for large $N$
 exponentially decrease and turn into zero for the infinite
 lattice.
 Finite sums (\ref{2.20}), (\ref{2.21}) transform into series,
summation over phase volume in (\ref{2.22}) is substituted by
integration and in the issue the form factor representations on
the cylinder turn into form factor representations on the infinite
lattice \cite{Palmev}, \cite{yamada1}. For any finite $N$ both
expansions -- over even $n$ (\ref{2.20}) and over odd $n$
(\ref{2.21}) -- are valid in both ferromagnetic $(s>1)$ and
paramagnetic $(s<1)$ regions.
 We remind that we started from the determinant (\ref{2.18})
 of a $|x|\times|x|$ matrix. The number of terms in its formal
 definition rapidly increases when $x$ grows.
 However, the form factor representations (\ref{2.20})--(\ref{2.23})
 are the finite sums for any fixed $N$, and the number of terms
 does not depend on $|x|$. This gives a unique opportunity to
 verify (\ref{2.20})--(\ref{2.23}) by means of comparing with
 the results of transfer matrix calculations for $N-$rows Ising chains.
 For fixed $N$ the dimension of corresponding transfer matrix
 is equal to $2^N\times2^N$.
 One can find analytically all eigenvectors and eigenvalues
 if $N$ is not too large. We have successfully performed such check analytically for
 $N=2,3,4$ and numerically -- for
$N=5,6$.

\section{Correlation function $\langle\sigma(0,0)\sigma(x,y)\rangle$}
 The rigorous derivation of the form factor
 representation on the cylinder was performed
 in \cite{Bugrij} only for the spins displaced along the cylinder
 axis.
 We have not yet succeeded in generalization of the method for
 arbitrary disposition of correlating spins (Fig. 1b). Meanwhile,
 the evaluation of the momentum representation of correlation
 function
\begin{equation}
\label{3.1}
\widetilde{G}(\pv)=\sum_{\rv}\e^{i\pv\rv}\langle\sigma(0)\sigma(\rv)
\rangle,
\end{equation}
 or the susceptibility (which is connected with
$\widetilde{G}(\pv=0)$) requires explicit dependence on both
components of the vector $\rv$. Form factor representations
~(\ref{2.20})--(\ref{2.23}) have a transparent physical content.
 This allows to make reasonable assumptions for corresponding
 generalizations. The above mentioned possibility of independent check
  allows to eliminate wrong hypotheses and to make correct choice.
  In principle, when $y$-component of the vector $\rv$ is not
  zero, all quantities in (\ref{2.20})--(\ref{2.23}) could change their
  form. Corresponding expressions for free bosons and fermions on the
  lattice prompt one of the simplest generalizations -- just the substitution
   $$ \e^{-|x|\gamma(q)}\to\e^{-|x|\gamma(q)-iyq}. $$
 Really amazing that it is enough. If instead of
$g_n(x)$~(\ref{2.22}) one uses the expression
\begin{equation}
g_n(\rv)=\frac{\e^{-n/\Lambda}}{n!N^n}{\sum_{[q]}}^{(b)}\prod_{j=1}^n
\left(\frac{\e^{-|x|\gamma_j-iyq_j-\eta_j}}{\sinh\gamma_j}\right)F_n^2[q],
\quad g_0=1, \label{3.3}
\end{equation} then correlation functions~(\ref{2.20}) and (\ref{2.21})
exactly coincide with transfer matrix results for $N=2,3,4$ in the
whole range of variables $x$, $y$, $K$. Numerical calculations
confirm this for $N=5,6$ also. The validity of ~(\ref{3.3}) is out
of doubts and we hope that the known answer will simplify the
problem of its rigorous derivation.

Let us illustrate the matter by the example of $N=3$. The
expansion (\ref{2.20})--(\ref{2.21}) are very similar to the
representation of the correlation function in terms of eigenvalues
of the transfer matrix
\begin{equation}
\langle\sigma(0)\sigma(\rv)\rangle=a_1(y)(\lambda_1/\lambda_0)^{|x|}+
a_2(y)(\lambda_2/\lambda_0)^{|x|}+\cdots,\label{3.4}
\end{equation}
 where $\lambda_0$ is the largest eigenvalue, coefficients
$a_j(y)$ are given by some bilinear combinations of eigenvectors.
To reduce, for example, (\ref{2.21}) to (\ref{3.4}), we use the
following expressions for cylindrical parameters $\xi_T$,
$\Lambda^{-1}$, $\eta(q)$
 \begin{eqnarray}\label{3.5}\Lambda^{-1}&=&\frac12
 \biggl({\sum_q}^{(f)}\gamma(q)
 -{\sum_q}^{(b)}\gamma(q)\biggr),\\
 \label{3.6}
 \e^{-\Lambda^{-1}-\eta(q_i)}&=&\frac{{\prod\limits_q}^{(b)}
\sinh\left( \frac{\gamma(q)+\gamma(q_i)}{2}\right)}
{{\prod\limits_q}^{(f)}\sinh\left(
\frac{\gamma(q)+\gamma(q_i)}{2}\right)},\\
\label{3.7}\xi_T^4&=&\frac{{\prod\limits_q}^{(b)}{\prod\limits_p}^{(f)}\sinh^2\left(
\frac{\gamma(q)+\gamma(p)}{2}\right)}
{{\prod\limits_q}^{(b)}{\prod\limits_p}^{(b)}\sinh\left(
\frac{\gamma(q)+\gamma(p)}{2}\right){\prod\limits_q}^{(f)}{\prod\limits_p}^{(f)}
\sinh\left( \frac{\gamma(q)+\gamma(p)}{2}\right)}. \end{eqnarray}
 One can derive these expressions from (\ref{2.25})--(\ref{2.27})
 by the substitution of the integration variable $z=\e^{iq}$ and
 computing the residues after proper squeezing the integration contours.

For $N=3$ we have from (\ref{3.5})--(\ref{3.7}) and (\ref{2.24})
\begin{eqnarray}\label{3.8}
\Lambda^{-1}&=&\frac{1}{2}\bigl[
 \gamma(\pi)+2\gamma({\pi}/{3})-\gamma(0)-2\gamma(
 {2\pi}/{3})
 \bigr],\\
 \xi\xi_{T}&=&\frac
 {\sinh\frac{\gamma(0)+\gamma(\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(2\pi/3)}{2}
  \sinh^{2}\frac{\gamma(2\pi/3)+\gamma(\pi/3)}{2}}
 {\sinh\frac{\gamma(0)+\gamma(2\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(\pi/3)}{2}
  \sinh\gamma(\pi/3)\sinh\gamma(2\pi/3)},\\
e^{-\Lambda^{-1}-\eta(q)}&=&
 \frac
 {\sinh\frac{\gamma(0)+\gamma(q)}{2}
  \sinh^{2}\frac{\gamma(2\pi/3)+\gamma(q)}{2}}
 {\sinh\frac{\gamma(\pi)+\gamma(q)}{2}
  \sinh^{2}\frac{\gamma(\pi/3)+\gamma(q)}{2}}.\label{3.10}
  \end{eqnarray}
 Finally,
\begin{eqnarray}\label{3.11}
\ln(\lambda_0/\lambda_1)&=&\Lambda^{-1}+\gamma(0),\\
\ln(\lambda_0/\lambda_2)&=&\Lambda^{-1}+\gamma(2\pi/3),\\
\ln(\lambda_0/\lambda_3)&=&\Lambda^{-1}+\gamma(0)+2\gamma(2\pi/3),
\end{eqnarray}
\begin{eqnarray}
\label{3.14}
 a_1(y)&=&\frac{1}{3} \frac
 {\sinh\frac{\gamma(0)+\gamma(2\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(2\pi/3)}{2}
  \sinh^{2}\frac{\gamma(2\pi/3)+\gamma(\pi/3)}{2}}
 {\sinh\frac{\gamma(0)+\gamma(\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(\pi/3)}{2}
  \sinh\gamma(\pi/3)\sinh\gamma(2\pi/3)},\\ a_2(y)&=&\frac{2}{3}\frac
 {\sinh\frac{\gamma(0)+\gamma(\pi/3)}{2}
  \sinh\frac{\gamma(0)+\gamma(\pi)}{2}}
 {\sinh\gamma(\pi/3)
  \sinh\frac{\gamma(\pi/3)+\gamma(\pi)}{2}} \cos
  (2\pi y/3),
\\ \label{3.16} a_3(y)&=&\frac{1}{64}\frac
  {1}
  {\sinh\frac{\gamma(0)+\gamma(\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(\pi/3)}{2}
  \sinh\frac{\gamma(0)+\gamma(2\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(2\pi/3)}{2}
    } \times\\
    \nonumber&& \times\frac{1}{\sinh\gamma(\pi/3)\sinh\gamma(2\pi/3)
  \sinh^{2}\frac{\gamma(\pi/3)+\gamma(2\pi/3)}{2}}.\end{eqnarray}

 The transfer matrix $2^3\times 2^3$ has 8 eigenvalues, some of them are
 equal. Besides that, some eigenvectors have zero components.
As result, the expression for the correlation function (\ref{3.4})
contains
 only three (not seven) independent terms.
 If we take into account the definition (\ref{2.15}), (\ref{2.16})
 of the function $\gamma(q)$ for particular values of quasimomentum
 $q=0,\ \pi/3,\ 2\pi/3,\
\pi$, we get exact correspondence between this three terms and
(\ref{3.11})--(\ref{3.16}).

\section{Momentum representation of the correlation\\ function}

Since we have the expression (\ref{3.3}) for $g_n(\rv)$, which
depends on both components of $\rv$, we can make the Fourier
transform. Let us write the momentum representation of (\ref{3.1})
in the form similar to (\ref{2.20})--(\ref{2.21})
\begin{equation}\label{4.1}
\widetilde{G}(\pv)=\xi\xi_T\sum_n\widetilde{g}_n(\pv),\end{equation}
\begin{equation}\label{4.2}
\widetilde{g}_n(\pv)=\sum_{\rv}\e^{-|x|/\Lambda}g_n(\rv)\e^{i\pv\rv},
\end{equation}
where
\begin{equation}
\sum_{\rv}=\sum_{x=-\infty}^\infty\sum_{y=1}^N.
\end{equation}
After performing the summation in (\ref{4.2}) we have
\begin{equation}\label{4.4}
\widetilde{g}_n(\pv)=\frac{\e^{n/\Lambda}}{n!N^{n-1}}
{\sum_{[q]}}^{(b)}\biggl(\prod_{j=1}^n\frac{\e^{-\eta_j}}{\sinh\gamma_j}
\biggr)\frac{\sinh\biggl(\Lambda^{-1}+\sum\limits_{j=1}^n\gamma_j
\biggr)F_n^2[q]}
{\cosh\biggl(\Lambda^{-1}+\sum\limits_{j=1}^n\gamma_j\biggr)-\cos
p_x}\delta\biggl(p_y-\sum_{j=1}^nq_j\biggr).
\end{equation}
The component $p_x$ of the quasimomentum has a continuous spectrum
in the range $[-\pi,\pi]$, but the $p_y$ is discrete $$
p_y=\frac{2\pi l}{N},\qquad l=1,\,2\,\,\ldots\,N. $$ Corresponding
$\delta$-function in the right hand side of (\ref{4.4}) is
understood as the Kronecker symbol $$
\delta\biggl(p_y-\sum_{j=1}^nq_j\biggr)=\delta\biggl(l-\sum_{j=1}^nl_j
\biggr)\biggm|_{\displaystyle{{\rm{mod}}\,N}}\,. $$ The function
$\widetilde{g}_n(\pv)$ is periodic in $p_x$, $p_y$ with the period
$2\pi$. After inserting the ``unity''  $$
1=\int\limits^{\Lambda^{-1}+n\gamma(\pi)}_{\Lambda^{-1}+n\gamma(0)}d\omega
\,\delta\biggl(\Lambda^{-1}+\sum^n_{j=1}\gamma_j-\omega\biggr), $$
in the sum (\ref{4.4}) (here $\delta$ denotes Dirac
$\delta$-function ) and changing the order of integration we
obtain
\begin{equation}\label{4.8}
\widetilde{g}_n(\pv)=\int\limits^{\Lambda^{-1}+n\gamma(\pi)}
_{\Lambda^{-1}+n\gamma(0)}d\omega\frac{\sinh\omega}{\cosh\omega-\cos
p_x}\rho_n(\omega, p_y),
\end{equation}
\begin{equation}\label{4.9}
{\rho}_n(\omega,p_y)=\frac{\e^{-n/\Lambda}}{n!N^{n-1}}
{\sum_{[q]}}^{(b)}\biggl(\prod_{j=1}^n\frac{\e^{-\eta_j}}{\sinh\gamma_j}
\biggr)F^2_n[q]\delta\biggl(\Lambda^{-1}+\sum_{j=1}^n\gamma_j-
\omega\biggr)\delta\biggl(p_y-\sum_{j=1}^nq_j\biggr).\end{equation}
On the infinite lattice in the scaling limit the rotational
symmetry is restored and (\ref{4.8}), (\ref{4.9}) turn into
classical Lehmann representation in the quantum field theory.

\section{Magnetic susceptibility}
 On the $M\times N$ square lattice with equal horizontal and vertical
 coupling parameters the partition function $Z$ depends on four
 variables
\begin{equation}
Z=Z(K,h,N,M)=\sum_{[\sigma]}\e^{-\beta
H[\sigma]+h\sum\limits_{\rv}\sigma(\rv)},
\end{equation}
where dimensionless parameter $h=\beta \HC$,
 $\HC$ -- magnetic field.
The specific magnetization $\M$ and magnetic susceptibility $\chi$
can be expressed through field derivatives of the partition
function
\begin{equation}
\M(K,h,N,M)=\frac{1}{MN}\frac{\partial\ln Z}{\partial
h}=\langle\sigma\rangle,
\end{equation}
\begin{equation}\label{5.3}
\beta^{-1}\chi(K,h,N,M)=\frac{\partial\M}{\partial
h}=\sum_{\rv}\biggl(\langle\sigma(0)\sigma(\rv)\rangle-\langle
\sigma\rangle^2\biggr).
\end{equation}
The magnetization at $h=0$ and finite $M$, $N$ turns into zero due
to $Z_2$-symmetry of the Ising model. This holds even when one of
the dimensions is set infinite. In the last case, when, for
example, $M\to\infty$, $N=\const$, 2D Ising model transforms into
1D chain with $N$ rows, for which a spontaneous symmetry breaking
is impossible. The susceptibility can be easily computed from
(\ref{4.1})--(\ref{4.4})
   \begin{eqnarray}\label{5.4}
\chi&=&\chi_0+ \sum_{l=1}^{[N/2]}\chi_{2l}\ {\text{\hspace{2.3cm}
 for\ \ }} \gamma(0)=\mu,\\\label{5.5}
\beta^{-1}\chi_0&=&\xi\xi_TN\coth(1/2\Lambda),\\\label{5.6}
\chi&=&\sum_{l=0}^{[(N-1)/2]}\chi_{2l+1}\ {\text{\hspace{2.3cm}
for\ \ }}\gamma(0)=-\mu,\\\label{5.7}
\beta^{-1}\chi_n&=&\frac{\e^{-n/\Lambda}}{
n!N^{n-1}}{\sum_{[q]}}^{(b)} \biggl(\prod_{i=1}^{n}
\frac{\e^{-\eta_i}}{\sinh \gamma_i}
\biggr)F_n^2[q]\coth\biggl[\frac12\biggl(\Lambda^{-1}+\sum_{i=1}^n
\gamma_i\biggr)\biggr]\delta\biggl(\sum_{i=1}^n q_i \biggr).
\end{eqnarray}
In paramagnetic region $(s<1)$ the expression (\ref{5.6}) admits
the limit $N\to\infty$ and turns into the susceptibility on the
infinite lattice. However, in the ferromagnetic region $(s>1)$ one
can make the limit $N\to\infty$ only for the quantity $\chi_F$
\begin{equation}\label{5.8}
\chi_F=\chi-\chi_0=\sum_{l=1}^{\infty}\chi_{2l},\end{equation}
which reproduces well-known zero-field ferromagnetic
susceptibility of the Ising model in thermodynamic limit. For
large but finite $N$ the main contribution to the susceptibility
is given by the term $\chi_0$
\begin{equation}
\beta^{-1}\chi_0\simeq2\xi N\Lambda\simeq\frac{\sqrt{\pi}\xi
N^{3/2}}{\sqrt{\sinh|\mu|}}\e^{N|\mu|},\label{5.9}
\end{equation}
which exponentially increases with the growth of the size of the
cylinder base. It follows from (\ref{5.9}) that the larger $N$ --
the smaller field $ \delta h\sim\e^{-N|\mu|}$ is needed to order
all spins on the lattice.

 Unfortunately, the exact solution for the partition function of
 the Ising model in external field is not known. However, the very fact
 of the appearance of spontaneous magnetization can be deduced
 from the analysis of high- and low-temperature expansions.
 The rigorous definition of spontaneous magnetization is given by
 the following order of limits according to the Bogolyubov concept
 of quasiaverages
\begin{equation}
\M_0(K)=\lim_{h\to0}\bigl[\lim_{M,N\to\infty}\M(K,h,N,M)\bigr].
\end{equation}
 However, if we conjecture the decreasing of correlations at large
 distances and the possibility of interchanging of corresponding
 limits, we can find the exact solution for the squared
 spontaneous magnetization. It is equal to spin-spin correlation
 function (\ref{2.24}) with infinite distance between correlating spins
\begin{equation}\label{5.12}
\M^2_0(K)=\lim_{|\rv|\to\infty}\langle\sigma(0)\sigma(\rv)\rangle=
\langle\sigma(0)\rangle\langle\sigma(\infty)\rangle=
\langle\sigma\rangle^2=\xi.
\end{equation}
Meanwhile, the sums over lattice of each summand in the right hand
side of (\ref{5.3}) do not converge in the thermodynamic limit.
Therefore, the substitution of $\M^2(K,0,\infty,\infty)$ by the
limiting value of correlation function (which equals $\xi$) under
the (infinite) sum in the last step of the limits $h\to0$,
$M,N\to\infty$ requires not only (\ref{5.12}), but also the
existence of the limit
\begin{equation}\label{5.13}
\lim_{h\to0}\bigl\{\lim_{M,N\to\infty}MN\bigl[\M^2(K,h,M,N)
-\xi\bigr]\bigr\}=f(K),
\end{equation}
and, moreover,
\begin{equation}\label{5.14}
 f(K)=0.
 \end{equation}
The explicit dependence of the correlation function on the size
$N$, namely, the exponential tending of cylindrical parameters to
their limiting values (\ref{2.28})--(\ref{2.30}), can be viewed as
an argument in favor of the equalities (\ref{5.13}), (\ref{5.14}).

The behaviour of correlation function at large distances in the
ferromagnetic region is mainly defined by the first term in the
expansion (\ref{2.20}). Note that it does not depend on
$y$-projection of $\rv$
\begin{equation}
G_0(|\rv|)=\xi\xi_T\e^{-|x|/\Lambda}.
\end{equation}
 Therefore, the distance $\sim\Lambda$, for which spins are strongly
 correlated, rapidly increases (cf (\ref{2.29})) with the growth of $N$.
 Physically it means that for ``ferromagnetic'' temperatures the cylinder
 is grained into ``domains'' of size $\sim\Lambda$ with nonzero
 magnetization, the magnetization of the whole infinite cylinder
 is being equal to zero. It is clear that the squared spontaneous
 magnetization would be more naturally defined by the value of the
 correlation function at large distances $|\rv|=R(N)$,
 which do not exceed the size of the domain
 $$ N\ll R(N)\ll\Lambda.$$
 It follows from (\ref{2.29}) that for sufficiently large $N$
these inequalities can be satisfied. In accordance with this, the
sum over $x$ with infinite limits in the definition of the
thermodynamic limit of susceptibility (\ref{5.3}) has to be
substituted by the sum with the limits that do not exceed the size
of the domain. In this case the condition
 $$
\sum_{x=-R}^R\sum_{y=1}^N[G_0(|\rv|)-G_0(R)]\simeq\xi
NR^2/\Lambda\mathop{\to}_{N\to\infty}0,$$ can be treated as a
formal substantiation of the definition (\ref{5.8}) of
susceptibility in the ferromagnetic phase. We can now estimate the
``parameter of thermodynamic cutting'' $R(N)$ $$
R(N)\ll\sqrt{\Lambda/N\xi}\simeq\e^{N|\mu|/2}[\pi/(2N\sinh|\mu|)]^{1/4}.
$$ We suppose that these estimations slightly clarify the physical
content of the formal thermodynamic limit procedure.

\section{Singularity structure}
 The initial expression (\ref{2.3}) for the partition function of
 the Ising model is a polynomial in
 $s$, and the solution (\ref{2.12}) is the factorized form of this
 polynomial.
 It provides an example of the mechanism of Lee-Yang ``zeros''
 \cite{Lee}, which stipulates the appearance of critical singularities
 in the thermodynamic limit. The roots of the polynomial (\ref{2.12})
are located on the unit circle $|s|=1$ in the complex $s$ plane.
For finite $M$ and $N$ the zero on the real axis $s=1$ does not
appear, since the fermionic spectrum does not contain the value of
quasimomentum $q_x=q_y=0$. When one of the dimensions increases
then zeros are concentrated on the circle $|s|=1$, forming a dense
set. In the limit $M\to\infty$, $N=\const$ they are transformed
into finite number (which equals $N$) of the root type
branchpoints, located on the circle $|s|=1$. To make sure of this,
one has to use the representation (\ref{2.14}) and definition
(\ref{2.15}), (\ref{2.16}) of the function $\gamma(q)$. These
branchpoints, in turn, form a dense set with the growth of $N$,
but in the limit $N\to\infty$ they are transformed into four
isolated logarithmic branchpoints $s=\pm1,\,\pm i$. As result, the
specific heat in the thermodynamic limit acquires the logarithmic
divergence $\sim\ln|1-s|$. It is worth noticing that the specific
heat is expressed through the same function in both ferromagnetic
and paramagnetic regions of $s$ contrary to the susceptibility.

One would think that the similar picture holds for susceptibility.
Indeed, the initial expression (\ref{2.4}) for the correlation
function for finite $M$ and $N$ is a ratio of polynomials in $s$.
The formation of the singularities of the partition function,
which stands in the denominator, we have just briefly described.
Unfortunately, the polynomial in the numerator cannot be written
in such simple factorized form. Nevertheless, our form factor
representation for $M\to\infty$ and finite $N$ shows that
correlation function has a finite number of root branchpoints on
the circle $|s|=1$. Their number is doubled in comparison with the
case of partition function, since the expressions
(\ref{2.20})--(\ref{2.23}), (\ref{3.3}) contain functions
$\gamma(q)$ (\ref{2.15}), corresponding to both bosonic and
fermionic values of quasimomentum. The susceptibility on the
cylinder is given by the infinite sum of correlation functions and
this can lead to the appearance of additional singularities. One
can show, however, that these singularities do not appear on the
first sheet of the Riemann surface.

 As an example, let us write down the susceptibility
$\chi$ (\ref{5.4}) for $N=3$, using the expressions
(\ref{3.14})--(\ref{3.16}) and representations
(\ref{3.8})--(\ref{3.10}) for cylindrical parameters
 \begin{eqnarray}\nonumber\beta^{-1}\chi&=&\frac
 {\sinh\frac{\gamma(0)+\gamma(2\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(2\pi/3)}{2}
  \sinh^{2}\frac{\gamma(2\pi/3)+\gamma(\pi/3)}{2}}
 {\sinh\frac{\gamma(0)+\gamma(\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(\pi/3)}{2}
  \sinh\gamma(\pi/3)\sinh\gamma(2\pi/3)}\:
  \coth\left(\tfrac{\Lambda^{-1}+\gamma(0)}{2}\right)+ \\
&& +\frac{1}{64}\frac{1}
  {\sinh\frac{\gamma(0)+\gamma(\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(\pi/3)}{2}
  \sinh\frac{\gamma(0)+\gamma(2\pi/3)}{2}
  \sinh\frac{\gamma(\pi)+\gamma(2\pi/3)}{2}
    } \times\label{6.1}\\
    \nonumber&& \times\frac{1}
     {\sinh\gamma(\pi/3)\sinh\gamma(2\pi/3)
  \sinh^{2}\frac{\gamma(\pi/3)+\gamma(2\pi/3)}{2}}
  \,\coth\left(\tfrac{\Lambda^{-1}+\gamma(0)+2\gamma(2\pi/3)}{2}\right).
 \end{eqnarray}
 The singularities in $s$ could appear due to zero denominator in (\ref{6.1}).
 It is easily seen, however, that corresponding factors
 $$\sinh\frac{\gamma(q)+\gamma(q')}{2}=(\cos q'-\cos q)/\sinh\frac
 {\gamma(q)-\gamma(q')}{2}$$
 for $q\neq q'$ do not equal zero. It can be also shown that on the first
 sheet of the Riemann surface (which is defined by the condition of
 positivity of $\gamma(q)$, treated as functions of $s$, for real $s>0$)
 the arguments of cotangents in
 (\ref{6.1}) also do not equal zero: these factors appear as the result
 of summation over coordinate $x$. Therefore, the complete set of
 the singularities of susceptibility is exhausted by the
 branchpoints contained in functions
 \begin{equation}
 \e^{\gamma(q)}=\biggl[\sqrt{\frac12(s+s^{-1})+\sin^2\frac q2}+
\sqrt{\frac12(s+s^{-1})-\cos^2\frac q2}\biggr]^2\label{6.2}
 \end{equation}
For each value of quasimomentum $q\neq0, \pi$ the function
(\ref{6.2}) has four branchpoints. If we denote them by
$s_c=|s_c|\e^{\pm i\varphi_c}$, then
\begin{equation}
|s_c|=1, \quad
\cos\varphi_c=\left\{\begin{array}{r}\cos^2q/2\\-\sin^2q/2\end{array}
\right.\,.\label{6.3}
\end{equation}
It is seen from (\ref{6.2}), that for $q=0, \pi$ there exist only
two branchpoints $s_c=\pm i$. One can now calculate that for any
fixed $N$ the whole number of singularities is equal to $4N-2$,
and all singularities are located on the unit circle $|s|=1$. We
represent the corresponding picture for $N=3$ in the Fig. 2.
\begin{figure}[h] \begin{center}
 \includegraphics[height=50mm,keepaspectratio=true]
 {Fig2-ms.eps}
 \caption{The location of the singularities of susceptibility
 $\chi$
 in the complex plane $s=\sinh 2\beta J$ for $N=3$.}
 \end{center} \end{figure}
  We do not discuss the limit $N\to\infty$, when the singularities on
  the circle $|s|=1$ form a dense set.
  This problem was seriously analyzed in
\cite{Nickel1}--\cite{Nickel2} and the authors assume (not prove)
that the singularities form a natural boundary $|s|=1$ for the
susceptibility.

 \vspace{1cm}
 We are indebted to V. N. Shadura for helpful discussions
 on the matters reflected in the paper.
 We wish to thank Professor B. M. McCoy for useful
 comments on the form factor representation of the correlation function
 on the cylinder and for bringing the problem of the singularities
 of susceptibility to our attention.

 This work was partly supported by INTAS program through grant
INTAS-97-1312.

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  \end{document}

