Double Potts chain and exact results for some two-dimensional models
A closed-form exact analytical solution for the $q$-state Potts model on a ladder $2\times\infty$ with arbitrary two-, three-, and four-site interactions in a unit cell is presented. Using the obtained solution it is shown that the finite-size internal energy equation [6] yields an accurate value of the critical temperature for the triangular Potts lattice with three-site interactions in alternate triangular faces. It is argued that the above equation is exact at least for self-dual models on isotropic lattices.
cond-mat0111401.tex

\documentstyle{article}

\begin{document}

\title{\bf Double Potts chain and exact results for some
two-dimensional models}

\author{ M.~A.~Yurishchev
\thanks{
On leave from Vasilsursk Laboratory,
Radiophysical Research Institute,
606263 Vasilsursk, Nizhny Novgorod Region, Russia
}
\thanks{E-mail: yur@itp.ac.ru}
}
\medskip
\date{\sl
The Abdus Salam International Centre for Theoretical Physics,
P.O.~Box 586, 34100 Trieste , Italy\\
}

\maketitle

\begin{abstract}
A closed-form exact analytical solution for the $q$-state Potts model
on a ladder $2\times\infty$ with arbitrary two-, three-, and four-site
interactions in a unit cell is presented.
Using the obtained solution it is shown that the finite-size
internal energy equation [6]
yields an accurate value of the critical temperature for the
triangular Potts lattice with three-site interactions in alternate
triangular faces.
It is argued that the above equation is exact at least for
self-dual models on isotropic lattices.
\end{abstract}

PACS: 05.50.+q, 05.70.Jk, 75.10.Hk

%----------------------------------------------------------------------

\newpage

\section{Introduction}
\label{sec:Intro}
The methods allowing to extract information about a
multi-dimensional system from solutions of their lower dimensional
counterparts play an important role in statistical physics.
One of the most well-known examples of this kind is
the finite-size scaling approach \cite{B83,P90}.

There are cases which evoke particular interest when the critical
properties of a system experiencing a phase transition can be
determined {\em exactly\/} from data only for its subsystems.
For instance, for the Ising strips the intersection point of the
locus of partition
function zeros in a complex temperature plane with the real
positive axis yields the exact value of the critical temperature
for the two-dimensional Ising model \cite{WTB87}.
Exact critical temperatures for the $S=1/2$ Ising models on
square, triangular, honeycomb, and centered square (Union Jack)
anisotropic lattices are obtained by using strip clusters when
an effective field is applied to one side of a strip only
\cite{LS92}.
Another exotic way of estimating the critical point of the square-lattice
Ising model was proposed in \cite{BS84}.
The authors of this paper showed that, in the quasi-diagonal
form of a transfer matrix of finite width strip, all
coefficients of the characteristic equation for the sub-block
containing the largest eigenvalue have extremum located precisely
at the exact value of phase transition temperature of infinite
lattice.

In the present paper we concentrate our attention on the method
to calculate the critical temperature, proposed by Wosiek \cite{W94}
(see also \cite{BW94,P95,BBSW95,SLS97,S98,S98a}).
The author \cite{W94} introduced  a maximum criterion
for the ratio of moments of the transfer matrix, and obtained the
following equation for determining the position of critical point in
a $d$-dimensional system:
\begin{equation}
   \label{eq:uu}
   u_1(K_c) = u_2(K_c) .
\end{equation}
Here $u_1$ and $u_2$ are the internal energies of $(d-1)$-dimensional
and two coupled $(d-1)$-dimensional subsystems, respectively, and
$K_c$ is the critical coupling (normalized inverse critical
temperature) of the $d$-dimensional system.

It is remarkable that at $d=2$ Eq.~(\ref{eq:uu}), shown in
\cite{W94}, yields the exact value of $K_c$ for the isotropic square
and triangular Ising lattices, as well as for the three-site Potts
model on the square lattice also with isotropic interactions.
Later on some other models were added to the list.
Now it includes another isotropic Baxter model (two square Ising
lattices coupled by four-particle interactions), the Baxter-Wu model
(triangular lattice with three-site interactions of Ising spins)
\cite{SLS97}, and $q$-state Potts model on the isotropic square
lattice with arbitrary value of $q$ \cite{S98a}.
Physical nature of Eq.~(\ref{eq:uu}) may be elucidated
when it yields either (a) an exact solution, or (b)
approximate estimation, or (c) does not give any solution at all
for a given model.
The more such examples are found, the better we understand its
sense.

For a two-dimensional system, Eq.~(\ref{eq:uu}) connects
the internal energies of infinitely long linear and double
chains.
Therefore, to obtain a possibility to test Eq.~(\ref{eq:uu}) rigorously,
it is necessary to have analytical solutions for such subsystems.

In the second section of the present paper, we give an exact
analytical solution for the two-chain Potts strip with a large
number of independent parameters.
As a special case, it contains a solution for the linear Potts chain.

Our solution for the double Potts chain enables us to cover all earlier
revealed cases in which Eq.~(\ref{eq:uu}) reproduces exactly the
critical temperatures for the two-dimensional models of Ising,
Baxter-Wu, and Potts.
Besides, we discover (section \ref{sec:TPL}) a new model for which
Eq.~(\ref{eq:uu}) yields the exact result.
This is the $q$-state Potts model on the triangular lattice with pure
three-site interactions in half of the triangular faces \cite{BTA78}.

In the fourth section of this paper a discussion of results is
presented.
In particular, it is shown that duality is a {\em sufficient\/}
condition for the validity of Eq.~(\ref{eq:uu}) for the isotropic
spin lattices.

Finally, in section 5, we summarize the results obtained in the work.

%---------------------------------------------------------------------

\section{Solution of double $q$-state Potts chain with
${\bf S}_q$ symmetry}
\label{sec:SDPC}

Consider a two-chain (ladder) lattice with spin variables
$\sigma_l^i$ attached to its sites ($i=1,2$ is the chain index,
and $l=1,2,3,\ldots$ labels the sites in the longitudinal direction
of the ladder); the spin variables take the values $1,2,\ldots,q$.

Let us write the Hamiltonian of a system in the form
\begin{equation}
   \label{eq:H}
   {\cal H} = -\sum_lH(\sigma_l^1,\sigma_l^2;
   \sigma_{l+1}^1,\sigma_{l+1}^2) .
\end{equation}
The locality of interactions in the above Hamiltonian allows one
to introduce the transfer matrix $V$ with elements
\begin{equation}
   \label{eq:V}
   \langle\sigma_1,\sigma_2|V|\sigma_1^{'},\sigma_2^{'}\rangle  =
   \exp[H(\sigma_1,\sigma_2; \sigma_1^{'},\sigma_2^{'})/k_BT]
\end{equation}
($T$ is the temperature and $k_B$ is Boltzmann's constant) and
reduce the problem of calculating the free energy density $f$ of
infinitely long strip to a search for the largest eigenvalue
$\lambda_1$ of the matrix $V$:
\begin{equation}
   \label{eq:f}
   f = \frac{1}{2}\ln\lambda_1 .
\end{equation}

The transfer matrix (\ref{eq:V}) has the size $q^2\times q^2$.
It is real and, moreover, all its elements are positive, but
the matrix is, generally speaking, non-symmetrical
($V_{ij}\neq V_{ji}$).

To solve the eigenvalue problem for the transfer matrix, we shall
use a group-theoretical approach (in this connection see, e.g.,
Ref.~\cite{Yu94} where such an approach was applied to a
quasidiagonalization of a transfer matrix of Ising model on
parallelepipeds $L\times L\times\infty$).
In order to obtain a solution for the two-leg spin ladder
(in which we are particularly interested) in a most general form,
we will go in reverse order.
Namely, we first select a symmetry group in the space
$|\sigma_1,\sigma_2\rangle$, that would enable us to quasidiagonalize
the transfer matrix up to sub-blocks secular equations which
can be solved analytically; only then we expand the
Hamiltonian density $H$ into a series on invariants of the
symmetry group.

Let us take a model which is invariant, for example, under
transformations of the symmetric group ${\bf S}_q$ of degree $q$.
For the Potts model this means, as a matter of fact, that we are
dealing with a system in the zero external field.
But fortunately the field is not required to test
Eq.~(\ref{eq:uu}).

It is known (see, for example, \cite{BN82}) that the largest eigenvalue of
a transfer matrix is located in the sub-block of the identity irreducible
representation (IR).
In accordance with group theory, the basis
vectors $\psi_i$ of the identity IR can be obtained  acting by
the permutation operators of a group ${\bf S}_q$ successively
on the orths $|1,1\rangle$, $|1,2\rangle$, $\ldots$\,,
$|q,q\rangle$.
Acting by elements of symmetric group first on the orth
$|1,1\rangle$ and then on the orth $|1,2\rangle$, we find that
the two linear combinations obtained involve all orths.
The normalized  basis vectors are given by
\begin{equation}
   \label{eq:psi}
   \psi_1 = \frac{1}{\sqrt{q}}\sum_{i=1}^q|i,i\rangle,\qquad
   \psi_2 = \frac{1}{\sqrt{q(q-1)}}\sum^q_{i,j=1}{}^{^{\!\prime}}
   |i,j\rangle
\end{equation}
(the prime at the second sum indicates that the terms with $i=j$ are
omitted).
Hence, the sub-block of identity IR has sizes 2 by 2
and, therefore, its eigenvalues (one of which is $\lambda_1$) can
easily be obtained by solving an algebraic equation only of second
degree.
Notice that if one takes the group ${\bf S}_q\times{\bf C}_s$
(${\bf C}_s$ is the group of mirror reflections in the plane going
between the chains of the two-leg ladder), the sub-block corresponding
to the identity IR will again have the size $2\times2$ and therefore
this symmetry, which only reduces the number of independent
parameters in a Hamiltonian, does not justify itself in the given
case.

We now represent the Hamiltonian (\ref{eq:H}) in the form of a sum
of terms which are invariant under transformations of the group
${\bf S}_q$:
\begin{eqnarray}
   \label{eq:HSq}
   & &
   {\cal H} = -\sum_l[
   J_1\delta_{\sigma^1_l\sigma^1_{l+1}} +
   J_2\delta_{\sigma^2_l\sigma^2_{l+1}} +
   J_0\delta_{\sigma^1_l\sigma^2_l} +
   J^{'}\delta_{\sigma^1_l\sigma^2_{l+1}} +
   J^{''}\delta_{\sigma^2_l\sigma^1_{l+1}}\nonumber\\
   & &\qquad
   + J_3\delta_{\sigma^1_l\sigma^2_l\sigma^1_{l+1}} +
   J_3^{'}\delta_{\sigma^1_l\sigma^1_{l+1}\sigma^2_{l+1}} +
   \tilde J_3\delta_{\sigma^1_l\sigma^2_l\sigma^2_{l+1}} +
   \tilde J_3^{'}\delta_{\sigma^2_l\sigma^1_{l+1}\sigma^2_{l+1}}\nonumber\\
   & &\qquad
   + J_4\delta_{\sigma^1_l\sigma^2_l\sigma^1_{l+1}\sigma^2_{l+1}}].
\end{eqnarray}
The Kronecker symbols entering here are defined as follows:
\begin{equation}
   \label{eq:Kr}
   \delta_{\sigma_1 \cdots\,\sigma_k} = \cases{
   1,&if $\sigma_1=\ldots =\sigma_k$\cr
   0,& otherwise} .
\end{equation}
The structure of two-site couplings in the Hamiltonian (\ref{eq:HSq})
is shown in Fig.~1.
Matrix elements of the original transfer matrix are written as
\begin{eqnarray}
   \label{eq:VSq}
   & &
   \langle\sigma_1,\sigma_2|V|\sigma_1^{'},\sigma_2^{'}\rangle  =
   \exp[
   K_1\delta_{\sigma_1\sigma_1^{'}} +
   K_2\delta_{\sigma_2\sigma_2^{'}} +
   {1\over 2}K_0(\delta_{\sigma_1\sigma_2} +
   \delta_{\sigma_1^{'}\sigma_2^{'}})\nonumber\\
   & &\qquad
   + K^{'}\delta_{\sigma_1\sigma_2^{'}} +
   K^{''}\delta_{\sigma_2\sigma_1^{'}} +
   K_3\delta_{\sigma_1\sigma_2}\delta_{\sigma_1\sigma_1^{'}} +
   K_3{'}\delta_{\sigma_1\sigma_1^{'}}\delta_{\sigma_1\sigma_2^{'}}\nonumber\\
   & &\qquad
   + \tilde K\delta_{\sigma_1\sigma_2}\delta_{\sigma_1\sigma_2^{'}} +
   \tilde K^{'}\delta_{\sigma_2\sigma_2^{'}}\delta_{\sigma_2\sigma_1^{'}} +
   K_4\delta_{\sigma_1\sigma_1^{'}}\delta_{\sigma_1\sigma_2}
   \delta_{\sigma_1\sigma_2^{'}}],
\end{eqnarray}
where $K_0=J_0/k_BT$, $K_1=J_1/k_BT$, $K_2=J_2/k_BT$, $K^{'}=J^{'}/k_BT$,
$K^{''}=J^{''}/k_BT$, $K_3=J_3/k_BT$, $K_3^{'}=J_3^{'}/k_BT$,
$\tilde K_3=\tilde J_3/k_BT$, $\tilde K_3^{'}=\tilde J_3^{'}/k_BT$, and
$K_4=J_4/k_BT$.

Using expressions (\ref{eq:psi}) and (\ref{eq:VSq}), we calculate
the matrix elements $Q_{ij}=\psi_i^{+}V\psi_j$ of the sub-block
corresponding to the identity IR:
\begin{eqnarray}
   \label{eq:Qij}
   & &
   Q_{11}=[q-1+\exp(K_1+K_2+K^{'}+K^{''}+K_3+K_3^{'}+\tilde K_3
   +\tilde K_3^{'}+K_4)]e^{K_0},\nonumber\\
   & &
   Q_{12}=(q-1)^{1/2}[q-2+\exp(K_1+K^{''}+K_3)+
   \exp(K_2+K^{'}+\tilde K_3)]e^{{1\over2}K_0},\nonumber\\
   & &
   Q_{21}=(q-1)^{1/2}[q-2+\exp(K_1+K^{'}+K_3^{'})+
   \exp(K_2+K^{''}+\tilde K_3^{'})]e^{{1\over2}K_0},\nonumber\\
   & &
   Q_{22}=(q-2)(q-3+e^{K_1}+e^{K_2}+e^{K^{'}}+e^{K^{''}})+
   \exp(K_1+K_2)+\exp(K^{'}+K^{''}).\nonumber\\
\end{eqnarray}
As a result, we find that the largest eigenvalue of the transfer
matrix of the double $q$-state Potts chain with the Hamiltonian
(\ref{eq:HSq}) reads
\begin{equation}
   \label{eq:L1}
   \lambda_1^{(2)} = {1\over2}(Q_{11}+Q_{22})+[{1\over4}(Q_{11}-
   Q_{22})^2+(q-1)Ae^{K_0}]^{1/2},
\end{equation}
where
\begin{eqnarray}
   \label{eq:A}
   & &
   A= [q-2+\exp(K_1+K^{''}+K_3)+ \exp(K_2+K^{'}+\tilde K_3)]
   [q-2\nonumber\\
   & &\qquad
   +\exp(K_1+K^{'}+K_3^{'})+
   \exp(K_2+K^{''}+\tilde K_3^{'})].
\end{eqnarray}

Those variants of the double Potts chains solved earlier
\cite{WTB87,S98a,S00,CS00,CS00a} correspond to a particular
choice of the interaction constants.
Setting $J_0=J^{'} (=J)$ with all other interaction
constants vanishing, we arrive at the solution for the linear
Potts chain \cite{P52}:
\begin{equation}
   \label{eq:L1D}
   \lambda_1^{(1)}(K) = e^K+q-1.
\end{equation}

%----------------------------------------------------------------------

\section{Triangular Potts lattice with three-site interactions on
alternate triangle faces}
\label{sec:TPL}

A large number of independent parameters in the model, solved in the
previous section, enables us to test Eq.~(\ref{eq:uu}) for a wide
class of two-dimensional spin systems.

In addition to the cases listed in the Introduction, in which
Eq.~(\ref{eq:uu}) is satisfied exactly, let us consider the Potts
model on a triangular lattice with three-site interactions in each
up-triangle (Fig.~2).
The position of the critical point in this model was found with both
three- and two-site interactions \cite{BTA78}.
However, it is known \cite{S98a} that for the triangle lattice with
pair couplings, Eq.~(\ref{eq:uu}) yields the exact result only for
the Ising case ($q=2$).
Therefore we discuss the model with purely three-site
interactions.
For this case,
\begin{equation}
   \label{eq:Kc}
   K_c = \ln(1+q).
\end{equation}
Let us show that this value satisfies Eq.~(\ref{eq:uu}) by subsystems in
the shape of strips with a periodic boundary condition in the
transverse direction.

The internal energy of one-dimensional subsystem is
\begin{equation}
   \label{eq:u1}
   u_1(K)\equiv\frac{\partial f_1}{\partial K} =
   [(q - 1)e^{-K} + 1]^{-1}.
\end{equation}
Substituting $K_3=\tilde K_3$ with all other interaction
constants vanishing, from (\ref{eq:f}), (\ref{eq:L1}), and (\ref{eq:A})
we obtain an expression for the free energy density of
double Potts chain:
\begin{eqnarray}
   \label{eq:f2}
   & &
   f_2(K) = {1\over 2}\ln[\![{1\over2}(e^{2K}+q^2-1)+
   [{1\over4}(e^{2K}-(q-1)^2)^2
   + q(q-1)(2e^K+\nonumber\\
   & &\qquad
   q-2)]^{1/2}]\!].
\end{eqnarray}
The internal energy $u_2(K)=\partial f_2/\partial K$.
Differentiating Eq.~(\ref{eq:f2}) with respect to $K$ one finds
an expression for $u_2(K)$.

An analysis shows that the dependences $u_1(K)$ and $u_2(K)$
have a crossing point which lies exactly at
$K=K_c=\ln(1+q)$ both for integer and non-integer $q$.
The internal energy of a system at critical point is
equal to
\begin{equation}
   \label{eq:uoo}
   u_{\infty}(K_c) = u_1(K_c) = u_2(K_c)
   = {1\over2}(1 + q^{-1}).
\end{equation}
Thus, using solutions only for the linear and double Potts chains,
Eq.~(\ref{eq:uu}) has enabled us to extract
the exact value of $K_c$ for the bulk two-dimensional Potts model
on a triangular lattice alternating faces which interact
by three-site forces.

%----------------------------------------------------------------------

\section{Discussion}

In the paper \cite{P95} Eq.~(\ref{eq:uu}) was extended up to
the form
\begin{equation}
   \label{eq:uLuL1}
   u_L(K_c) = u_{L'}(K_c)\qquad (L, L^{'} = 1, 2, 3,\,\ldots\,).
\end{equation}
Here $u_L$ is the internal energy per site of $L$ coupled
$(d-1)$-dimensional subsystems.
In the two-dimensional case $L$ denotes the width of a strip.

The validity of condition (\ref{eq:uLuL1}) for arbitrary $L$ and
$L^{'}$ means the absence of a ``singular'' (i.e. depending on $L$)
part of the internal energy density at critical point:
\begin{equation}
   \label{eq:uLKc}
   u_L(K_c) = const\quad {\rm upon}\quad L.
\end{equation}
In other words, the amplitudes of all finite-size corrections to the
critical internal energy of a system $u_{\infty}(K_c)$ are equal
to zero.

For the square isotropic Ising lattice, the derivative of the inverse
correlation length $\kappa_L(K)$ with respect to a temperature-like
variable $K$ has a similar property \cite{Yu00,Yu00a}:
\begin{equation}
   \label{eq:k1Kc}
   \frac{\partial\kappa_L}{\partial K}{\Biggr|}_{K=K_c} =
   \frac{\partial\kappa_{L^{'}}}{\partial K}{\Biggr|}_{K=K_c},
\end{equation}
i.e. $\partial\kappa_L/\partial K|_c$ does not depend on $L$.
This property has enabled us to determine exactly the value of
the thermal critical exponent $y_t\,(=1)$ for this model using only
the finite-size data \cite{Yu00,Yu00a}.

Equations (\ref{eq:uu}) and (\ref{eq:uLuL1}) are valid for the
{\em ferromagnetic\/} isotropic square Potts lattices.
These models are self-dual and the critical coupling for them
(in the anisotropic case) is determined from the condition
\begin{equation}
   \label{eq:Kcf}
   (e^{K_x} - 1)(e^{K_y} - 1) = q.
\end{equation}
For the {\em antiferromagnetic\/} square-lattice Potts model
the criticality condition is \cite{B82a}
\begin{equation}
   \label{eq:Kcaf}
   (e^{K_x} + 1)(e^{K_y} + 1) = 4 - q,
\end{equation}
where both $K_x < 0$ and $K_y < 0$.
We performed a verification and found that in the antiferromagnetic
case the curves $u_1(K)$ and $u_2(K)$ do not have any self-crossing
point and therefore Eq.~(\ref{eq:uu}) does not lead to the
exact value which follows from Eq.~(\ref{eq:Kcaf}) or even to any
approximate estimate for the critical point.

It is not difficult to show that if a model is self-dual and by this
the dual point coincides with the original one, then
Eqs.~(\ref{eq:uu}) and (\ref{eq:uLuL1}) are valid.

Indeed, consider for instance the Ising model on the isotropic
square lattice $L\times N$ with toroidal boundary conditions.
The partition function of such a system has a fundamental
property: it is invariant (up to multiplicative factor
exponentially depending on $L$) under the duality transformation
(see \cite{B82}):
\begin{equation}
   \label{eq:ZLN}
   Z_{L,N}(K^{*}) = (\sinh2K)^{-LM}Z_{L,M}(K),
\end{equation}
where $K$ and $K^{*}$ are related by the condition
\begin{equation}
   \label{eq:thK}
   \tanh K^{*} = e^{-2K}.
\end{equation}
(We here used another normalization of the exchanged constant in
the Ising model, namely $J_{\rm Potts}=2J_{\rm Ising}$.)
In the limit of an infinitely long strip ($N\to\infty$),
Eq.~(\ref{eq:ZLN}) transforms to the duality condition for the
largest eigenvalue:
\begin{equation}
   \label{eq:L1L}
   \lambda_1^{(L)}(K^{*}) = (\sinh2K)^{-L}\lambda_1^{(L)}(K).
\end{equation}
It follows from here that the values of the normalized internal
energy in dually conjugated points ($K$ and $K^*$) are connected
by the relation
\begin{equation}
   \label{eq:uLK}
   u_L(K^{*})\frac{\partial K^{*}}{\partial K} =
   u_L(K) - 2u_0(K)
\end{equation}
in which the additive term $u_0\ (=\coth 2K)$
{\em does not depend on\/} $L$.
Another important feature, this time related to isotropy of a
lattice, is that the dually conjugated points $K$ and $K^{*}$
confluent into one point at criticality:
\begin{equation}
   \label{eq:KK}
   K^{*} = K = K_c.
\end{equation}
Using condition (\ref{eq:thK}) we find that at the critical point
$K_c=(1/2)\ln(1+\sqrt2)$ the derivative
$\partial K^{*}/\partial K\bigr|_c=-1$.
Consequently
\begin{equation}
   \label{eq:uLu0Kc}
   u_L(K_c) = u_0(K_c) = \sqrt2.
\end{equation}
Thus, the critical internal energy per site $u_L(K_c)$ of
an Ising cylinder with isotropic square cells obeys the
condition (\ref{eq:uLKc}) for all $L=1,2,\ldots$.
That, in turn, leads to the validity of Eqs.~(\ref{eq:uu}) and
(\ref{eq:uLuL1}).

In an analogous way, Eqs.~(\ref{eq:uu}) and (\ref{eq:uLuL1})
can be derived for other isotropic spin models partition
functions which satisfy a functional equation like
\begin{equation}
   \label{eq:ZLg}
   Z_L(K^{*}) = [g(K)]^LZ_L(K).
\end{equation}

In those cases when the model is self-dual but the critical
manifold is a line or a surface (as, e.g., for anisotropic
lattices), Eqs.~(\ref{eq:uu}) and (\ref{eq:uLuL1}) no longer
hold.

This is not difficult to prove if we consider again a
two-dimensional Ising model.
For the anisotropic square lattice the duality condition
now looks as
\begin{equation}
   \label{eq:L1La}
   \lambda_1^{(L)}(K_x^{*}, K_y^{*}) =
   (\sinh2K_x\sinh2K_y)^{-L/2}\lambda_1^{(L)}(K_x, K_y)
\end{equation}
with
\begin{equation}
   \label{eq:thKxy}
   \tanh K_x^{*} = e^{-2K_y}\quad {\rm and}\quad
   \tanh K_y^{*} = e^{-2K_x}.
\end{equation}
From this it follows that on the critical line
\begin{equation}
   \label{eq:shKxy}
   \sinh2K_x \sinh2K_y = 1
\end{equation}
a condition (\ref{eq:L1La}) relates the values of the free
energy in {\em distinct\/} (dually conjugated) points
($K_x,K_y$) and ($K_y,K_x$):
\begin{equation}
   \label{eq:fLKxy}
   f_L(K_x, K_y) = f_L(K_y, K_x) + {1\over2}\ln(\sinh2K_x\sinh2K_y).
\end{equation}
This circumstance prevents fulfillment of the validity of
Eqs.~(\ref{eq:uu}) and (\ref{eq:uLuL1}) which identify
the internal energies at the same point.

The critical internal energy density of a strip $L\times\infty$
cut out from {\em anisotropic} lattice depends on the size $L$.
This is easy to verify if, using the results of the section 2,
one calculates the values $u_1(K_c)$ and $u_2(K_c)$ for the
anisotropic Ising and Potts lattices.

On the other hand, we can establish the same property if we take the
Onsager solution \cite{O44} for the two-dimensional Ising
model.
The dominant eigenvalue of a transfer matrix of cylinder
$L\times\infty$ with spatially anisotropic interactions
is equal to
\begin{equation}
   \label{eq:lam1}
   \lambda_1^{(L)}(K;\alpha) = (2\sinh 2K)^{L/2}\exp [(\gamma_1 +
   \gamma_3 + \ldots + \gamma_{2L-1})/2],
\end{equation}
where $\alpha=J_y/J_x$ is the lattice anisotropy parameter and
$\gamma_r$ are positive solutions of the equations
\begin{equation}
   \label{eq:chgam}
   \cosh\gamma_r = \cosh2\alpha K\coth 2K -
   \frac{\sinh2\alpha K}{\sinh2K} \cos\Bigg(\frac{\pi r}{L}\Bigg).
\end{equation}
From this, for the internal energy per site, we obtain
\begin{equation}
   \label{eq:uLKa}
   u_L(K;\alpha) = \coth2K + \frac{1}{2L}\Bigg(
   \frac{\partial\gamma_1}{\partial K} +
   \frac{\partial\gamma_3}{\partial K} +
   \ldots +
   \frac{\partial\gamma_{2L-1}}{\partial K}\Bigg).
\end{equation}
The dependences $\gamma_r(K)$ have a smooth extremum
(minimum) which lies in the isotropic case ($\alpha=1$)
exactly at $K=K_c$ and therefore
\begin{equation}
   \label{eq:gamK}
   \frac{\partial\gamma_r}{\partial K}\Bigg|_{K=K_c} = 0\qquad
   (r\neq0).
\end{equation}
As a result, the second term in Eq.~(\ref{eq:uLKa}) disappears
and the critical internal energy ceases to depend on $L$.
When $\alpha\neq1$ then Eq.~(\ref{eq:gamK}) does not have a
place and $u_L(K; \alpha)$ depends on the strip width in
a complicated way.
This explains the failures of exact calculations of $K_c$
from Eq.~(\ref{eq:uu})
in the anisotropic Ising lattice \cite{SLS97}.

Ending the section we note that, in spite of Eqs.~(\ref{eq:uu})
and (\ref{eq:uLuL1}), Eq.~(\ref{eq:k1Kc}) cannot be
deduced from a dual invariance of system.

%----------------------------------------------------------------------

\section{Conclusions}

Using a group-theoretical approach we obtained the
exact analytical solution for the double Potts chain with the
Hamiltonian (\ref{eq:HSq}).
The solution gives a possibility to examine Eq.~(\ref{eq:uu})
for a large number of models both with Ising ($q=2$) and arbitrary
Potts spins (including non-integer $q$).
The validity of Eq.~(\ref{eq:uu}) for the
triangular Potts lattice with pure three-site interactions in
alternate triangular faces was established.

In this paper it was also shown that Eqs.~(\ref{eq:uu}) and
(\ref{eq:uLuL1}) are a consequence of a dual symmetry
of models for which the critical point coincides with its
dual image.

As far as the author knows, the inverse theorem has not been
proved.
Duality plus isotropy or, more correctly, self-conjugation of a
critical point are not {\em necessary\/} conditions for
Eq.~(\ref{eq:uu}).
Therefore, generally speaking, there can exist systems which are
not invariant under dual transformation or a combination of
dual and star-triangular ones, but for which all amplitudes of
finite-size corrections to the critical internal energy (or to
some other quantity) are equal to zero.

%----------------------------------------------------------------------

\section*{Acknowledgements}

The author thanks A.~A.~Belavin, A.~A.~Nersesyan, and A.~M.~Sterlin
for useful discussions and comments.
I am also grateful to the Abdus Salam International Centre for Theoretical
Physics (Trieste, Italy) for kind hospitality where
this work was finished.
The research presented in this paper is supported in part by the
grants RFBR No.~99-02-16472 and CRDF No.~RP1-2254.

%----------------------------------------------------------------------

\newpage

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(1983), Vol.~8, p.~145.

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\end{thebibliography}

%----------------------------------------------------------------------

\newpage

\section*{Figure captions}\ \\

Fig.~1.\
Geometry of two-site couplings in the double $q$-state Potts chain
with ${\bf S}_q$ symmetry.\\

Fig.~2.\
Fragment of Potts lattice with three-site interactions in
alternate triangular faces (shaded).

%----------------------------------------------------------------------

\end{document}
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%%EndProlog
%%BeginSetup
%%Feature: *Resolution 600dpi
TeXDict begin
%%PaperSize: A4

%%EndSetup
%%Page: 1 1
1 0 bop 474 3199 a @beginspecial 0 @llx 0 @lly 617 @urx
321 @ury 3401 @rwi @setspecial
%%BeginDocument: fig2.eps
/MyAppDict 100 dict dup begin def
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {0 setgray} bind def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
newpath 0 321 moveto 0 0 lineto 617 0 lineto 617 321 lineto closepath clip newpath
% Fill background color
0 0 moveto 617 0 lineto 617 321 lineto 0 321 lineto
closepath 1.00 1.00 1.00 setrgbcolor fill

-11.0 415.0 translate
1 -1 scale
.9 .9 scale % to make patterns same scale as in xfig

% This junk string is used by the show operators
/PATsstr 1 string def
/PATawidthshow {        % cx cy cchar rx ry string
  % Loop over each character in the string
  {  % cx cy cchar rx ry char
    % Show the character
    dup                         % cx cy cchar rx ry char char
    PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char)
    false charpath              % cx cy cchar rx ry char
    /clip load PATdraw
    % Move past the character (charpath modified the
    % current point)
    currentpoint                        % cx cy cchar rx ry char x y
    newpath
    moveto                      % cx cy cchar rx ry char
    % Reposition by cx,cy if the character in the string is cchar
    3 index eq {                        % cx cy cchar rx ry
      4 index 4 index rmoveto
    } if
    % Reposition all characters by rx ry
    2 copy rmoveto              % cx cy cchar rx ry
  } forall
  pop pop pop pop pop           % -
  currentpoint
  newpath
  moveto
} bind def
/PATcg {
  7 dict dup begin
    /lw currentlinewidth def
    /lc currentlinecap def
    /lj currentlinejoin def
    /ml currentmiterlimit def
    /ds [ currentdash ] def
    /cc [ currentrgbcolor ] def
    /cm matrix currentmatrix def
  end
} bind def
% PATdraw - calculates the boundaries of the object and
% fills it with the current pattern
/PATdraw {                      % proc
  save exch
    PATpcalc                    % proc nw nh px py
    5 -1 roll exec              % nw nh px py
    newpath
    PATfill                     % -
  restore
} bind def
% PATfill - performs the tiling for the shape
/PATfill { % nw nh px py PATfill -
  PATDict /CurrentPattern get dup begin
    setfont
    % Set the coordinate system to Pattern Space
    PatternGState PATsg
    % Set the color for uncolored pattezns
    PaintType 2 eq { PATDict /PColor get PATsc } if
    % Create the string for showing
    3 index string              % nw nh px py str
    % Loop for each of the pattern sources
    0 1 Multi 1 sub {           % nw nh px py str source
        % Move to the starting location
        3 index 3 index         % nw nh px py str source px py
        moveto                  % nw nh px py str source
        % For multiple sources, set the appropriate color
        Multi 1 ne { dup PC exch get PATsc } if
        % Set the appropriate string for the source
        0 1 7 index 1 sub { 2 index exch 2 index put } for pop
        % Loop over the number of vertical cells
        3 index                 % nw nh px py str nh
        {                       % nw nh px py str
          currentpoint          % nw nh px py str cx cy
          2 index show          % nw nh px py str cx cy
          YStep add moveto      % nw nh px py str
        } repeat                % nw nh px py str
    } for
    5 { pop } repeat
  end
} bind def

% PATkshow - kshow with the current pattezn
/PATkshow {                     % proc string
  exch bind                     % string proc
  1 index 0 get                 % string proc char
  % Loop over all but the last character in the string
  0 1 4 index length 2 sub {
                                % string proc char idx
    % Find the n+1th character in the string
    3 index exch 1 add get      % string proe char char+1
    exch 2 copy                 % strinq proc char+1 char char+1 char
    % Now show the nth character
    PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr)
    false charpath              % string proc char+1 char char+1
    /clip load PATdraw
    % Move past the character (charpath modified the current point)
    currentpoint newpath moveto
    % Execute the user proc (should consume char and char+1)
    mark 3 1 roll               % string proc char+1 mark char char+1
    4 index exec                % string proc char+1 mark...
    cleartomark                 % string proc char+1
  } for
  % Now display the last character
  PATsstr dup 0 4 -1 roll put   % string proc (char+1)
  false charpath                % string proc
  /clip load PATdraw
  neewath
  pop pop                       % -
} bind def
% PATmp - the makepattern equivalent
/PATmp {                        % patdict patmtx PATmp patinstance
  exch dup length 7 add         % We will add 6 new entries plus 1 FID
  dict copy                     % Create a new dictionary
  begin
    % Matrix to install when painting the pattern
    TilingType PATtcalc
    /PatternGState PATcg def
    PatternGState /cm 3 -1 roll put
    % Check for multi pattern sources (Level 1 fast color patterns)
    currentdict /Multi known not { /Multi 1 def } if
    % Font dictionary definitions
    /FontType 3 def
    % Create a dummy encoding vector
    /Encoding 256 array def
    3 string 0 1 255 {
      Encoding exch dup 3 index cvs cvn put } for pop
    /FontMatrix matrix def
    /FontBBox BBox def
    /BuildChar {
        mark 3 1 roll           % mark dict char
        exch begin
        Multi 1 ne {PaintData exch get}{pop} ifelse  % mark [paintdata]
          PaintType 2 eq Multi 1 ne or
          { XStep 0 FontBBox aload pop setcachedevice }
          { XStep 0 setcharwidth } ifelse
          currentdict           % mark [paintdata] dict
          /PaintProc load       % mark [paintdata] dict paintproc
        end
        gsave
          false PATredef exec true PATredef
        grestore
        cleartomark             % -
    } bind def
    currentdict
  end                           % newdict
  /foo exch                     % /foo newlict
  definefont                    % newfont
} bind def
% PATpcalc - calculates the starting point and width/height
% of the tile fill for the shape
/PATpcalc {     % - PATpcalc nw nh px py
  PATDict /CurrentPattern get begin
    gsave
        % Set up the coordinate system to Pattern Space
        % and lock down pattern
        PatternGState /cm get setmatrix
        BBox aload pop pop pop translate
        % Determine the bounding box of the shape
        pathbbox                        % llx lly urx ury
    grestore
    % Determine (nw, nh) the # of cells to paint width and height
    PatHeight div ceiling               % llx lly urx qh
    4 1 roll                            % qh llx lly urx
    PatWidth div ceiling                % qh llx lly qw
    4 1 roll                            % qw qh llx lly
    PatHeight div floor                 % qw qh llx ph
    4 1 roll                            % ph qw qh llx
    PatWidth div floor                  % ph qw qh pw
    4 1 roll                            % pw ph qw qh
    2 index sub cvi abs                 % pw ph qs qh-ph
    exch 3 index sub cvi abs exch       % pw ph nw=qw-pw nh=qh-ph
    % Determine the starting point of the pattern fill
    %(px, py)
    4 2 roll                            % nw nh pw ph
    PatHeight mul                       % nw nh pw py
    exch                                % nw nh py pw
    PatWidth mul exch                   % nw nh px py
  end
} bind def

% Save the original routines so that we can use them later on
/oldfill        /fill load def
/oldeofill      /eofill load def
/oldstroke      /stroke load def
/oldshow        /show load def
/oldashow       /ashow load def
/oldwidthshow   /widthshow load def
/oldawidthshow  /awidthshow load def
/oldkshow       /kshow load def

% These defs are necessary so that subsequent procs don't bind in
% the originals
/fill      { oldfill } bind def
/eofill    { oldeofill } bind def
/stroke    { oldstroke } bind def
/show      { oldshow } bind def
/ashow     { oldashow } bind def
/widthshow { oldwidthshow } bind def
/awidthshow { oldawidthshow } bind def
/kshow     { oldkshow } bind def
/PATredef {
  MyAppDict begin
    {
    /fill { /clip load PATdraw newpath } bind def
    /eofill { /eoclip load PATdraw newpath } bind def
    /stroke { PATstroke } bind def
    /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
    /ashow { 0 0 null 6 3 roll PATawidthshow }
    bind def
    /widthshow { 0 0 3 -1 roll PATawidthshow }
    bind def
    /awidthshow { PATawidthshow } bind def
    /kshow { PATkshow } bind def
  } {
    /fill   { oldfill } bind def
    /eofill { oldeofill } bind def
    /stroke { oldstroke } bind def
    /show   { oldshow } bind def
    /ashow  { oldashow } bind def
    /widthshow { oldwidthshow } bind def
    /awidthshow { oldawidthshow } bind def
    /kshow  { oldkshow } bind def
    } ifelse
  end
} bind def
false PATredef
% Conditionally define setcmykcolor if not available
/setcmykcolor where { pop } {
  /setcmykcolor {
    1 sub 4 1 roll
    3 {
        3 index add neg dup 0 lt { pop 0 } if 3 1 roll
    } repeat
    setrgbcolor - pop
  } bind def
} ifelse
/PATsc {                % colorarray
  aload length          % c1 ... cn length
    dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
  } ifelse } ifelse
} bind def
/PATsg {                % dict
  begin
    lw setlinewidth
    lc setlinecap
    lj setlinejoin
    ml setmiterlimit
    ds aload pop setdash
    cc aload pop setrgbcolor
    cm setmatrix
  end
} bind def

/PATDict 3 dict def
/PATsp {
  true PATredef
  PATDict begin
    /CurrentPattern exch def
    % If it's an uncolored pattern, save the color
    CurrentPattern /PaintType get 2 eq {
      /PColor exch def
    } if
    /CColor [ currentrgbcolor ] def
  end
} bind def
% PATstroke - stroke with the current pattern
/PATstroke {
  countdictstack
  save
  mark
  {
    currentpoint strokepath moveto
    PATpcalc                            % proc nw nh px py
    clip newpath PATfill
    } stopped {
        (*** PATstroke Warning: Path is too complex, stroking
          with gray) =
    cleartomark
    restore
    countdictstack exch sub dup 0 gt
        { { end } repeat } { pop } ifelse
    gsave 0.5 setgray oldstroke grestore
  } { pop restore pop } ifelse
  newpath
} bind def
/PATtcalc {             % modmtx tilingtype PATtcalc tilematrix
  % Note: tiling types 2 and 3 are not supported
  gsave
    exch concat                                 % tilingtype
    matrix currentmatrix exch                   % cmtx tilingtype
    % Tiling type 1 and 3: constant spacing
    2 ne {
        % Distort the pattern so that it occupies
        % an integral number of device pixels
        dup 4 get exch dup 5 get exch           % tx ty cmtx
        XStep 0 dtransform
        round exch round exch                   % tx ty cmtx dx.x dx.y
        XStep div exch XStep div exch           % tx ty cmtx a b
        0 YStep dtransform
        round exch round exch                   % tx ty cmtx a b dy.x dy.y
        YStep div exch YStep div exch           % tx ty cmtx a b c d
        7 -3 roll astore                        % { a b c d tx ty }
    } if
  grestore
} bind def
/PATusp {
  false PATredef
  PATDict begin
    CColor PATsc
  end
} bind def

% right30
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 32 16 true [ 32 0 0 -16 0 16 ]
        {<00030003000c000c0030003000c000c0030003000c000c00
        30003000c000c00000030003000c000c0030003000c000c0
        030003000c000c0030003000c000c000>}
     imagemask } bind
] def
/PaintProc {
        pop
        exec fill
} def
currentdict
end
/P2 exch def

% left45
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 32 32 true [ 32 0 0 -32 0 32 ]
        {<808080804040404020202020101010100808080804040404
        020202020101010180808080404040402020202010101010
        080808080404040402020202010101018080808040404040
        202020201010101008080808040404040202020201010101
        808080804040404020202020101010100808080804040404
        0202020201010101>}
     imagemask } bind
] def
/PaintProc {
        pop
        exec fill
} def
currentdict
end
/P4 exch def
1.1111 1.1111 scale %restore scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
10 setmiterlimit
 0.06000 0.06000 sc
% Polyline
7.500 slw
n 1425 1800 m 1425 4200 l 4200 4200 l 1425 1800 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 95.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 4200 1800 m 4200 4200 l 6975 4200 l 4200 1800 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 280.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 6975 1800 m 6975 4200 l 9750 4200 l 6975 1800 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 465.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 1425 1800 m 1425 4200 l 4200 4200 l 1425 1800 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 95.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 1425 4200 m 1425 6600 l 4200 6600 l 1425 4200 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 95.00 280.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 4200 4200 m 4200 6600 l 6975 6600 l 4200 4200 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 280.00 280.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 6975 4200 m 6975 6600 l 9750 6600 l 6975 4200 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 465.00 280.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 1425 4200 m 1425 6600 l 4200 6600 l 1425 4200 l
 cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P2 [16 0 0 -8 95.00 280.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 1050 1800 m
 10275 1800 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P4 [16 0 0 -16 70.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 1050 4200 m
 10275 4200 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P4 [16 0 0 -16 70.00 280.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 1050 6600 m
 10275 6600 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P4 [16 0 0 -16 70.00 440.00] PATmp PATsp ef gr PATusp gs col0 s gr
% Polyline
n 9750 1800 m
 9750 6600 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P4 [16 0 0 -16 650.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr
/Times-Roman ff 300.00 scf sf
975 6750 m
gs 1 -1 sc (1) dup sw pop neg 0 rm  col0 sh gr
/Times-Roman ff 300.00 scf sf
975 1950 m
gs 1 -1 sc (1) dup sw pop neg 0 rm  col0 sh gr
/Times-Roman ff 300.00 scf sf
975 4350 m
gs 1 -1 sc (2 \(1\)) dup sw pop neg 0 rm  col0 sh gr
$F2psEnd
rs
end
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f(some)h(t)m(w)m(o-dimensional)d(mo)s(dels")328 1120
y(Fig.)h(2.)p eop
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cond-mat0111401.extracted_bib
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