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\begin{document}


\articletitle[Field Theory of Scaling Lattice Models]
{Field theory of scaling \\
lattice models.\\
The Potts antiferromagnet\footnote{Talk given at the NATO Advanced Research 
Workshop on Statistical Field Theories, Como 18-23 June 2001}}

\author{Gesualdo Delfino}

%% affil, email, and abstract are optional
\affil{SISSA\\
via Beirut 2-4, 34014 Trieste\\
Italy}
\email{delfino@sissa.it}

%% optional, to supply a shorter version of the title for the running head:
%%\chaptitlerunninghead{}



\begin{abstract}
In contrast to what happens for ferromagnets, the lattice structure 
participates in a crucial way to determine existence and type of critical 
behaviour in antiferromagnetic systems. It is an interesting question to
investigate how the memory of the lattice survives in the field theory
describing a scaling antiferromagnet. We discuss this issue for the square 
lattice three-state Potts model, whose scaling limit as $T\rightarrow 0$ is
argued to be described exactly by the sine-Gordon field theory at a specific
value of the coupling. The solution 
of the scaling ferromagnetic case is recalled for comparison. The field theory
describing the crossover from antiferromagnetic to ferromagnetic behaviour
is also introduced.
\end{abstract}

\begin{keywords}
Field theory, statistical mechanics, antiferromagnets, Potts model, 
integrability
\end{keywords}

\section{Introduction}

The description of statistical systems nearby their phase transition points
is one of the most stimulating applications of quantum field theory. 
The typical problem one has to face in this context is that of 
{\em non-trivial} fixed points of the renormalisation group, i.e. of {\em 
strongly interacting} quantum field theories. It is then an important 
discovery of the last 
years that quantum field theory can actually be done non-perturbatively in two 
dimensions. Here, conformal \cite{BPZ} and integrable \cite{Taniguchi} field 
theories describe {\em exactly} the critical points and the scaling limits of 
statistical models, respectively. 

This fact has been exploited to solve the scaling ferromagnetic models in 
two dimensions directly in the continuum limit. 
Since scaling ferromagnets exhibit universal 
behaviour independent on lattice details, the recipe is\,: find the simplest 
integrable field theory with the internal symmetry characteristic of the 
given universality class. It is remarkable that the basic two-dimensional 
ferromagnets 
%(the Ising model in a 
%magnetic field \cite{Taniguchi}, the $q$-state Potts model \cite{qPotts}, 
%the $O(n)$ model \cite{Oenne}, just to make few examples) 
are integrable in the 
continuum limit (not on the lattice, usually) and can be solved in this way.

It should not be surprising that much less is known for
antiferromagnets. The spins in an antiferromagnet want to be in a state which
is different from that of their nearest neighbours. Hence, the number of such
neighbours, namely the lattice structure, participates in a crucial way to 
determine existence and type of critical behaviour. It follows that the 
phenomenology is much richer than in the ferromagnetic case. Nevertheless, if 
a critical point with infinite correlation length exists, there must be a 
quantum field theory describing the scaling limit. It is an intriguing 
question to find such a theory and to understand how it `remembers' about
the lattice structure. Once the lattice and internal symmetries have been 
disentangled in the field theory of the scaling antiferromagnet, another 
interesting point is that of breaking the first symmetry while preserving the 
second. The result should be a theory describing the crossover from
antiferromagnetic to ferromagnetic behaviour.

Once again, one expects that in two dimensions these issues can be investigated
in a precise, non-perturbative way. We will show that this is indeed the case 
for a non-trivial example, the three-state Potts 
model on the square lattice. In the next Section we recall the solution of 
the ferromagnetic case before turning to the antiferromagnet in Section 3 and
discussing the crossover between the two in Section 4.

\section{Scaling Limit of the Ferromagnetic Three-state Potts Model}

The three-state Potts model is defined by the Hamiltonian
\EQ
H=-J\sum_{\langle i,j\rangle}\delta_{s_i,s_j}\,\,,
\hspace{1cm}s_i=1,2,3
\label{lattice}
\EN
where $s_i$ denotes the spin located at the $i$-th site of a regular lattice 
and the sum is taken
over nearest neighbours. The model is characterised by the invariance under
global permutations of the values of the spin. The permutation group $S_3$ 
can be seen as the product of the group $Z_3$ of cyclic permutations times
a ``charge conjugation'' $C$. The elementary $Z_3$ transformation and 
charge conjugation act as follows on the complex spin variable 
$\sigma_j=e^{2i\pi s_j/3}$\,:
$$
Z_3\,:\hspace{.3cm}\sigma_j\rightarrow e^{2i\pi/3}\,\sigma_j\,,\hspace{1.5cm}
C\,:\hspace{.3cm}\sigma_j\rightarrow\sigma_j^*\,\,;
$$
of course they leave invariant the energy operator 
$\varepsilon_j=\sum_i\delta_{s_i,s_j}$.

The model exhibits ferromagnetic or antiferromagnetic behaviour depending on
the sign of the coupling $J$. 

In two dimensions, the ferromagnetic ($J>0$) model undergoes a second order
phase transition at a critical temperature $T_c$ \cite{Baxter}. It was shown 
in \cite{Dotsenko,FZ} that the critical point is described by the minimal model
of conformal field theory with central charge $C=4/5$. The spin 
operator $\sigma(x)$
and the energy operator $\varepsilon(x)$ correspond to the primary
conformal operators $\phi_{2,3}(x)$ and $\phi_{2,1}(x)$ with scaling dimensions
$X_\sigma=2/15$ and $X_\varepsilon=4/5$, respectively. As a consequence,
the scaling limit can be described by adding to the fixed point action 
${\cal A}_{C=4/5}$ the thermal perturbation in the form
\EQ
{\cal A}_F={\cal A}_{C=4/5}+\tau\int d^2x\,\varepsilon(x)\,,
\label{action}
\EN
with $\tau$ measuring the deviation from $T_c$. While the model (\ref{lattice})
is not solvable on the lattice away from $T=T_c$ \cite{Baxter}, the scaling
limit (\ref{action}) belongs to the large class of 
integrable quantum field theories discovered by A. Zamolodchikov 
\cite{Taniguchi}. 

Integrable quantum field theories can be solved exactly in the scattering
theory framework \cite{ZZ}. In fact, integrability (i.e. the existence of an 
infinite number of quantum integrals of motion) ensures the complete elasticity
and factorisation of the scattering processes and allows the determination
of the scattering amplitudes. Universality requires that the scaling limit
of the ferromagnetic three-state Potts model corresponds to the simplest
integrable scattering theory implementing the $S_3$ symmetry. The difference
between the high and low temperature phases is made by the nature of the 
excitations. 

At $T>T_c$ there is a single ground state and the simplest realisation of the 
symmetry is in terms of a doublet of charge conjugated particles $A$ and 
$\bar{A}$ of mass $m$ transforming under $S_3$ as the spin operators $\sigma$ 
and $\sigma^*$, respectively. The $Z_3$ symmetry in enforced by requiring 
the existence of the fusion process
\EQ
A\,A\rightarrow \bar{A}\,\,.
\label{fusion}
\EN
Factorisation of multiparticle processes implies that the full scattering 
matrix is determined by the three two-particle amplitudes depicted in Fig.\,1.
The last process turns out to be incompatible with factorisation and 
the property (\ref{fusion}) together, and the simplest solution with 
$S_3^F(\theta)=0$ is \cite{KS,Sasha}
\EQ
S_1^F(\theta)\,=\,S_2^F(i\pi-\theta)\,\,=\,\,
\frac{\sinh\frac12(\theta+\frac{2i\pi}{3})}
                           {\sinh\frac12(\theta-\frac{2i\pi}{3})}\,,
\EN
where $\theta$ parameterises the center of mass energy
$\sqrt{s}=2m\cosh(\theta/2)$.
The nonvanishing amplitudes are related by crossing symmetry while unitarity
follows from the fact that complex conjugation amounts to $\theta
\rightarrow -\theta$.
The pole at $\theta=2\pi i/3$ in the amplitude $S_1^F(\theta)$ corresponds to 
the bound state (\ref{fusion}). 


At $T<T_c$ there are three degenerate ground states and the excitations are 
``kinks'' $K_{j,j\pm 1}$ interpolating between the ground state $j=1,2,3$ 
and the ground state $j+1$ (mod 3). Their space-time trajctories draw domain
walls separating regions with different magnetisation. Due to invariance 
under permutations, there are only three inequivalent two-kink scattering 
amplitudes which are readily mapped into those of Fig.\,1 through the 
identifications $K_{j,j+1}\longleftrightarrow A$, 
$K_{j,j-1}\longleftrightarrow \bar{A}$. As a consequence the solution for the
scattering amplitudes is the same than at $T>T_c$. 
This is the way in which the high-low temperature duality of the Potts model
emerges in this context. The computation of the correlation functions in the 
two phases starting from the scattering solution can be found in \cite{DC}.

\begin{figure}
\centerline{
\psfig{figure=crossampl.ps}}
\caption{The scattering amplitudes $S_1$, $S_2$ and $S_3$.} 
\end{figure}


\section{The Square Lattice Antiferromagnet}
At $T=0$ the antiferromagnetic ($J<0$) three-state Potts model on the {\em 
square lattice} finds infinitely many ground states in which each spin
has a value different from that of its nearest neighbours. 
After transferring the labels from sites to faces, a configuration
in terms of arrows can be obtained through the following rule: if an 
observer in one face labelled $j$ looks across an edge to an adjacent face
labelled $j+1$ (mod 3), then put an arrow on this edge pointing to the 
observer's left; if the adjacent face is labelled $j-1$ (mod 3), point the 
arrow to the right. All sites of the resulting configuration 
satisfy the rule ``two arrows in, two arrows out'' which defines the 
six-vertex model. The latter is exactly solvable on the lattice \cite{Baxter}
and is known to be critical and equivalent at long distance to a free massless 
boson. 

This exact mapping of the square lattice antiferromagnet (\ref{lattice}) 
at zero-temperature onto a particular case\footnote{The one in which all 
vertices are equally weighted, the so called ice point.} of the six-vertex 
model has been known for long time and has been used to determine the scaling
dimensions of a number of relevant operators \cite{dNNS,PW,BH,SS}. These
are the staggered magnetisation 
$\Sigma_j=(-1)^{j_1+j_2}\,e^{2i\pi s_j/3}$ with dimension $1/6$, 
the uniform magnetisation $\sigma_j=e^{2i\pi s_j/3}$ with
dimension $2/3$, and the staggered polarisation\footnote{The primed sum 
indicates summation over the next nearest neighbours of $j$.} 
${\cal P}_j=(-1)^{j_1+j_2}\sum'_i(2\delta_{s_i,s_j}-1)$ with 
dimension $3/2$. Here and in the following we identify the $j$-th site of the
square lattice through a pair of integers $(j_1,j_2)$, and call even (odd)
sublattice the collection of the sites with $j_1+j_2$ even (odd). 

The model is not solved on the lattice at non-zero temperature and the 
issue of the approach to criticality (which involves first of all the 
determination of the scaling dimension of the energy operator 
$\varepsilon_j=\sum_i\delta_{s_i,s_j}$) has made the object of recent studies
\cite{FS,CJS,af}. The authors of \cite{CJS} exploited a mapping onto a height
model to study the lattice excitations at $T>0$ and explain the anomalous
corrections to scaling observed in simulations. Here we will follow
Ref. \cite{af} to show how the scaling limit as $T\rightarrow 0$ is in fact
described by an integrable quantum field theory.

Since the critical model is described by a gaussian fixed point, the 
expectation that the scaling limit corresponds to a massive integrable
field theory is very natural. In fact, under the (mild) assumption that
the thermal operator $\varepsilon_j$ gives in the continuum limit a single
{\em relevant} operator $\varepsilon(x)$, the action one is left with for the
scaling limit is that of the sine-Gordon model,
\EQ
{\cal A}_{AF}=\int d^2x\,\left(\frac12\,\partial_a\varphi\partial^a\varphi
-\mu\cos\beta\varphi\right)\,,
\label{sg}
\EN
which is integrable. Having said that, it remains to be understood how the 
action (\ref{sg}) actually describes the Potts antiferromagnet, namely where 
this action hides the relevant lattice and $S_3$ symmetries, how the Potts 
degrees of freedom
are expressed in terms of the bosonic field $\varphi$, and which value of 
$\beta$ determines the scaling dimension $X_\varepsilon=\beta^2/4\pi$ 
of the energy operator $\varepsilon=\cos\beta\varphi$.

To answer these questions we have to recall few facts about the operator 
content of the sine-Gordon model. At the gaussian fixed point ($\mu=0$)
the bosonic field decomposes into holomorphic and antiholomorphic parts as
$\varphi(x)=\phi(z)+\bar{\phi}(\bar{z})$, 
where $z=x_1+ix_2$ and $\bar{z}=x_1-ix_2$. The scaling operators 
$V_{p,\bar{p}}(x)=\exp{i[p\,\phi(z)+\bar{p}\bar{\phi}(\bar{z})]}$
have scaling dimension $X=(p^2+\bar{p}^2)/8\pi$, spin $s=(p^2-\bar{p}^2)/8\pi$,
and satisfy the gaussian operator product expansion
\EQ
V_{p_1,\bar{p}_1}(x)V_{p_2,\bar{p}_2}(0)=
z^{p_1p_2/{4\pi}}\,\bar{z}^{\bar{p}_1\bar{p}_2/{4\pi}}\,
V_{p_1+p_2,\bar{p}_1+\bar{p}_2}(0)+\ldots\,.
\label{ope}
\EN
This relation shows that taking $V_{p_1,\bar{p}_1}(x)$ around 
$V_{p_2,\bar{p}_2}(0)$ by sending $z\rightarrow ze^{2i\pi}$ and 
$\bar{z}\rightarrow \bar{z}e^{-2i\pi}$ produces a phase factor
$e^{2i\pi\gamma_{1,2}}$, where
$\gamma_{1,2}=(p_1p_2-\bar{p}_1\bar{p}_2)/(4\pi)$
is called index of mutual locality. If $\gamma_{1,2}$ is an integer the 
correlators $\langle ..V_{p_1,\bar{p}_1}(x)V_{p_2,\bar{p}_2}(0)..\rangle$ are
single valued as functions of $x$ and the two operators are said to be 
mutually local. Since $\gamma_{1,1}=2s$, the operators which are local with 
respect to themselves must have integer or half integer spin.

The operators of interest for the description of the statistical model
are scalar ($s=0$) and local with respect to the energy 
$\varepsilon=\cos\beta\varphi$. These requirements select
\bea
V_p&\equiv& V_{p,p}=\exp [ip\varphi]\,,
\label{Vp}\\
U_m &\equiv& V_{2\pi m/\beta,-2\pi m/\beta}=
\exp [2i\pi m\tilde{\varphi}/\beta]\,,\hspace{.3cm}m=\pm 1,\pm 2,..
\label{Um}
\eea
where $\tilde{\varphi}(x)\equiv\phi(z)-\bar{\phi}(\bar{z})$ is sometimes 
called the ``dual'' boson. A slightly more general analysis extended to the
spin 1/2 operators shows that the integer $m$ in (\ref{Um}) is in fact the
topologic charge that in the sine-Gordon model originates from the periodicity 
of the potential. Hence the operators $U_m(x)$ have charge $m$ while the 
operators $V_p(x)$ are neutral. This implies in particular that in the model
(\ref{sg})
\EQ
\langle V_p\rangle\neq 0\,,\hspace{1cm}\langle U_m\rangle=0\,\,.
\label{vevs}
\EN
Another point to be remarked is that the operators $U_m(x)$ with $|m|>3$ are 
always irrelevant ($X>2$) as long as the perturbation in (\ref{sg}) is
relevant (i.e. $\beta^2<8\pi$).

The lattice operators $\Sigma_j$, $\sigma_j$ and ${\cal P}_j$ are not 
invariant under the $S_3$ symmetry and/or the exchange of the even and odd
sublattices. Hence their continuum counterparts have to be sought among the 
$U_m(x)$. Comparison with the known scaling dimensions shows that the 
matching is complete provided we take $\beta=\sqrt{6\pi}$, 
what in turn implies $X_\varepsilon=3/2$. The operator identifications are 
summarised in Table\,1 from which the following correspondences can also be 
read:
\begin{center}
$Z_3$ charge =\,\,$m$\,(mod 3)

$C\,=$ complex conjugation

sublattice parity = $(-1)^m$\,\,.
\end{center}
\noindent
Hence we see that in the continuum limit both the $Z_3$ symmetry
and the lattice symmetry are ruled by the topologic charge $m$.

\begin{table}[ht]
\caption{Relevant operators on the lattice and their 
continuum counterparts in the sine-Gordon model.}       
\begin{tabular}{c c l c c}\sphline
 & Lattice definition & Continuum limit & $X$ & m \\ \sphline
$\Sigma$ & $(-1)^{j_1+j_2}\,\exp[2i\pi s_j/3]$ & $U_1=\exp[i\sqrt{2\pi/3}\,
\tilde{\varphi}]$ & $1/6$ & $1$ \\ 
$\sigma$ & $\exp [2i\pi s_j/3]$ & $U_{-2}=
\exp [-i\sqrt{8\pi/3}\,\tilde{\varphi}]$ & $2/3$ & $-2$ \\
${\cal P}$ & $(-1)^{j_1+j_2}\sum_i'(2\delta_{s_i,s_j}-1)$ &
$U_3+U_{-3}=\cos\sqrt{6\pi}\,\tilde{\varphi}$ & $3/2$ & $\pm 3$ \\ 
$\varepsilon$ & $\sum_i\delta_{s_i,s_j}$ & $V_{\sqrt{6\pi}}+V_{-\sqrt{6\pi}}=
\cos\sqrt{6\pi}\,\varphi$ & $3/2$ & $0$ \\
\sphline
\end{tabular}
\end{table}
                                 
Due to the integrability of the sine-Gordon model, also the scaling 
antiferromagnet admits an exact scattering description. This time the 
elementary excitations are the soliton $A$ and antisoliton $\bar{A}$ 
interpolating between adjacent sine-Gordon vacua. They carry topologic
charge 1 and $-1$, respectively, and then transform under the symmetries as 
the {\em staggered} magnetisation $\Sigma$ and $\Sigma^*$. This is what makes 
the difference with the ferromagnetic case at the level of the scattering 
theory: the sublattice parity (which plays no role in the ferromagnet) 
would now be violated by the fusion process (\ref{fusion}), which is therefore
forbidden. One consequence is that the last amplitude in Fig.\,1 is no longer 
forced to vanish. The three amplitudes are the sine-Gordon ones \cite{ZZ},
\bea
S_1^{AF}(\theta)= S_2^{AF}(i\pi-\theta)=
-\exp\left\{\int_0^\infty\frac{dx}{x}\frac{\sinh\frac{x}{2}\left(1
-\frac{\xi}{\pi}\right)}{i\sinh\frac{x\xi}{2\pi}\cosh\frac{x}{2}}
\sin\frac{\theta x}{\pi}\right\}\\
S_3^{AF}(\theta)= -\frac{\sinh\frac{i\pi^2}{\xi}}
               {\sinh\frac{\pi}{\xi}(\theta-i\pi)}\,\,S_1^{AF}(\theta)\,,
\hspace{4.8cm}
\eea
evaluated at the value $\xi=3\pi$ corresponding to $\beta=\sqrt{6\pi}$. This
value falls in the sine-Gordon
repulsive region in which the solitons do not form any bound state. In 
particular no asymptotic particle corresponding to the field $\varphi$ 
in (\ref{sg}) is present in the spectrum.

Correlation functions can be computed starting from the scattering theory
through the form factor approach (see \cite{af}). Here we only mention few 
straighforward predictions for the antiferromagnet dictated by the topologic 
charge of the operators. Defining the `exponential' correlation 
length $\xi_\Phi$ associated to an operator $\Phi(x)$ as
\EQ
\langle\Phi(x)\Phi^*(0)\rangle_{\mbox{connected}}\sim\exp(-|x|/\xi_\Phi)\,,\hspace{1cm}
|x|\rightarrow\infty\,,
\EN
then the following universal ratios should be observed in simulations as
$T\rightarrow 0$\,:
$\xi_\sigma/\xi_\Sigma=\xi_{\cal E}/\xi_\Sigma=1/2$, $\xi_{\cal P}/
\xi_\Sigma=1/3$.

\section{Crossover from Antiferromagnetic to Ferromagnetic Behaviour}
We see from Table\,1 that the field theory of the scaling antiferromagnet 
contains a {\em single} relevant operator with the symmetry properties 
required to break the sublattice symmetry while keeping the $S_3$ symmetry.
Therefore the action that should describe the crossover to ferromagnetic 
behaviour reads
\bea
{\cal A}_{cross} &=& {\cal A}_{AF}-\tilde{\mu}\int d^2x\,{\cal P}(x)\nonumber\\
         &=& {\cal A}_{C=1}-\int d^2x\,[\,
\mu\cos\sqrt{6\pi}\,\varphi+\tilde{\mu}\cos\sqrt{6\pi}\,\tilde{\varphi}\,]\,,
\label{cross}
\eea
and defines a one-parameter family of renormalisation group trajectories
(labelled by $\mu/\tilde{\mu}$)
flowing out of the gaussian ($C=1$) fixed point. Since ${\cal P}(x)$ has 
topologic charge $\pm 3$, it reintroduces in the theory the three-particle 
vertex (\ref{fusion}), so that the $Z_3$ symmetry is again manifest.

One of the trajectories described by (\ref{cross}) should flow into the 
ferromagnetic ($C=4/5$) fixed point in the infrared limit. This trajectory
marks a phase boundary across which a continous ordering phase transition 
takes place. The existence of such a transition can be argued as follows. 
When $\tilde{\mu}=0$ we are in the sine-Gordon model (\ref{sg}) in which 
the vacuum expectation value of the spin operator
$\sigma=\exp[-i\sqrt{8\pi/3}\,\tilde{\varphi}]$ (i.e. the spontaneous
magnetisation) vanishes according to (\ref{vevs}). It can be argued in the 
spirit of Ref.\,\cite{msg} that, since ${\cal P}(x)$ is local with respect 
to the solitons of the theory (\ref{sg}), no phase transition takes place
as soon as the perturbation is switched on, namely $\langle\sigma\rangle=0$
for $\tilde{\mu}\simeq 0$. On the other hand, we can perform similar 
considerations for the opposite limit ($\mu\simeq 0$) of the action
(\ref{cross}), where we are perturbing around a sine-Gordon model of the 
{\em dual} boson $\tilde{\varphi}(x)$. The operator $\sigma(x)$ has zero 
topologic charge with respect to this model and its vacuum expectation value
no longer vanishes, namely $\langle\sigma\rangle\neq 0$ for $\mu\simeq 0$.
These conclusions for the two limits are compatible if a phase transition 
takes place in between giving rise to the phase diagram of Fig.\,2.

\begin{figure}
\centerline{
\psfig{figure=crosspd.ps}}
\caption{Phase diagram associated to the action 
(\ref{cross}).}
\end{figure}


Generically, the theory (\ref{cross}) will not be integrable when both 
perturbations are switched on. There are reasons to expect, however, that
the massless trajectory connecting the two fixed points is integrable.
As a matter of fact, some years ago Fateev and Al. Zamolodchikov used the 
thermodynamic Bethe ansatz to provide evidence for an infinite series of 
integrable flows between the $Z_N$ parafermionic conformal field theories 
(with central charge $2(N-1)/(N+2)$) and the unitary minimal models ${\cal
M}_{N+1}$ (central charge $1-6/[(N+1)(N+2)]$) \cite{FalZ}. The case $N=4$ 
gives a flow between the fixed points with central charges $C=1$ and $C=4/5$ 
which should correspond to the one we are discussing in connection with the 
Potts model and the action (\ref{cross}). 

A relation between $Z_4$ symmetry and the Potts antiferromagnetic fixed point 
was suggested in Ref. \cite{Saleur}. It should be stressed, however, 
that this symmetry corresponds to an universality class which differs from 
that of the Potts antiferromagnet\footnote{The scaling dimensions of the 
spin and energy operators in the $Z_4$ model are 1/8 and 2/3, respectively 
\cite{FZ}. The possibility that the ``same'' quantum field theory 
accounts for different universality classes is well known (see e.g. Ref. 
\cite{rsos}, Section 6, for a discussion and references on the argument).},
and that it is explicitely broken in the off-critical actions (\ref{sg}) and 
(\ref{cross}). According to the discussion of this section,
the flow from $C=1$ to $C=4/5$ should be described by a factorised massless 
$S$-matrix exhibiting the $S_3$ symmetry characteristic of the 3-state Potts
model. To the best of our knowledge this $S$-matrix did not appear in the 
literature so far.

\section{Conclusion}
We have seen that both the ferromagnetic and (square lattice) antiferromagnetic
scaling limits of the three-state Potts model correspond to integrable 
quantum field theories, and that the massless crossover trajectory should also 
be integrable. It would be interesting to confirm this latter point by 
determining the exact massless $S$-matrix. Of course, one can think of 
extending the investigation of this paper to other antiferromagnets 
possessing a critical point.

We conclude by mentioning that actions of the type (\ref{cross})
containing both kinds of scaling operators (\ref{Vp}) and (\ref{Um}) are not
uncommon in the description of scaling lattice models. For example, the 
Ashkin-Teller model consists of two Ising models coupled through their 
energy terms (four spin interaction). As long as the two Ising models are 
kept at the same temperature the scaling limit is described 
by the 
action (\ref{sg}) with $\beta$ parameterising this time the line of fixed
points characteristic of the Ashkin-Teller model (see \cite{AT}). A 
temperature difference corresponds to the addition of the operator
$\cos[4\pi\tilde{\varphi}/\beta]$. It can be argued that also this action
admits a massless flow which in this case ends in the infrared limit
into an Ising fixed point with $C=1/2$.


\vspace{.3cm}
\noindent
{\bf Acknowledgments:} I thank J. Cardy and V. Fateev for interesting 
discussions.

\notes


\begin{chapthebibliography}{1}
\bibitem{BPZ} A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, {\em Nucl.
Phys.} {\bf B 241} (1984) 333.
\bibitem{Taniguchi} A.B. Zamolodchikov, {\em Adv. Stud. Pure Math.} {\bf 19}
(1989) 641; {\em Int. J. Mod. Phys.} {\bf A3} (1988) 743.
\bibitem{Baxter} R.J. Baxter, Exactly solved models of statistical
mechanics, Academic Press, London, 1982.
\bibitem{Dotsenko} Vl.S. Dotsenko, {\em Nucl. Phys.} {\bf B 235} (1984) 54.
\bibitem{FZ} V.A. Fateev and A.B. Zamolodchikov, {\em Sov. Phys. JEPT} 
{\bf 62} (1985) 215.
\bibitem{ZZ} A.B. Zamolodchikov and Al.B. Zamolodchikov, {\em Ann. Phys.} 
{\bf 120} (1979), 253.
\bibitem{KS} R. Koberle and J.A. Swieca, {\em Phys. Lett.} {\bf B 86}
(1979) 209.
\bibitem{Sasha} A.B. Zamolodchikov, {\em Int. J. Mod. Phys.} {\bf A 3}
(1988) 743.
\bibitem{DC} G. Delfino and J.L. Cardy, {\em Nucl. Phys.} {\bf B 519} (1998) 
551.
\bibitem{dNNS} M. den Nijs, M.P. Nightingale and M. Schick, {\em Phys.
Rev.} {\bf B 26} (1982) 2490.
\bibitem{PW} H. Park and M. Widom, {\em Phys. Rev. Lett.} {\bf 63} (1989)
1193.
\bibitem{BH} J.K. Burton Jr. and C.L. Henley, {\em J. Phys.} {\bf A 30}
(1997) 8385.
\bibitem{SS} J. Salas and A.D. Sokal, {\em J. Stat. Phys.} {\bf 92} (1998)
729.
\bibitem{FS} S.J. Ferreira and A.D. Sokal, {\em J. Stat. Phys.} {\bf 96}
(1999) 461.
\bibitem{CJS} J.L. Cardy, J.L. Jacobsen and A.D. Sokal, cond-mat/0101197, 
to appear in {\em J. Stat. Phys.}
\bibitem{af} G. Delfino, {\em J. Phys.} {\bf A 34} (2001) L311.
\bibitem{msg} G. Delfino and G. Mussardo, {\em Nucl. Phys.} {\bf B 516} 
(1998) 675.
\bibitem{FalZ} V.A. Fateev and Al.B. Zamolodchikov, {\em Phys. Lett.}
{\bf B 271} (1991) 91.
\bibitem{Saleur} H. Saleur, {\em Nucl. Phys.} {\bf B 360} (1991) 219.
\bibitem{rsos} G. Delfino, {\em Nucl. Phys.} {\bf B 583} (2000) 597.
\bibitem{AT} G. Delfino, {\em Phys. Lett.} {\bf B 450} (1999) 196.

\end{chapthebibliography}


\end{document}








































\section[All the Things that can be Done with Figure Captions]
{All the Things that can be Done\\ with Figure Captions}

Here are some examples of various kinds of figure captions
that can be use with this Kluwer style. They include the
normal \LaTeX\ \verb+\caption{}+ as well as many more possibilities
which you will see illustrated here.

\begin{figure}[ht]
\vskip.2in
\caption{Short caption.}
\end{figure}

\noindent
The following example shows a caption which includes an indexing command.
Notice that there is a \verb+\protect+ command before the \verb+\inx+.
This keeps \LaTeX\ from expanding the \verb+\inx+ command at
the wrong time.

\begin{figure}[ht]
\caption{\protect\inx{Oscillograph} for memory address access operations, showing 500 ps
address access time and $\alpha\beta\Gamma\Delta\sum_{123}^{345}$
\protect\inx{superimposed signals}%
\protect\inxx{address,superimposed
signals} of address access in 1 kbit
memory plane.}
\end{figure}

\noindent
Here is an example of a double caption; one figure with two
captions appearing side by side:

%% Double captions:
\begin{figure}[ht]
\sidebyside
{\caption{This caption will go on the left side of
the page. It is the initial caption of two side-by-side captions.}}
{\caption{This caption will go on the right side of
the page. It is the second of two side-by-side captions.}}
\end{figure}

\noindent
When you need a continued caption for a second figure that
uses the same number as the preceding one as a continuation
of the previous figure:

%% For continued caption. Same figure number used as for last caption.
\begin{figure}[ht]
\contcaption{This is a continued caption.}
\end{figure}
\inxx{captions,figure}

\noindent
When you want to make a narrow caption, you can use the
\verb=\narrowcaption= command.

%% To make narrow caption:
\begin{figure}[ht]
\narrowcaption{This is a narrow caption so that it can
be at the side of the illustration. This is a narrow caption.
This is a narrow caption. This is a narrow caption.}
\end{figure}

\noindent
You may also make a narrow continued caption as you see in
the following example.

%% To make narrow continued caption:
\begin{figure}[ht]
\narrowcontcaption{This is a narrow continued caption.
This is a narrow continued caption. This is a narrow continued caption.}
\end{figure}

\noindent
When you need to make a lettered caption, you may use the command\newline
\verb+\letteredcaption{}{}+. The first argument is
for the letter.

\begin{figure}[ht]
\letteredcaption{a}{Lettered caption.}
\end{figure}
\inxx{captions,lettered}


Notice that you can have lettered captions in the side by side
environment, which is one of the places that lettered captions
may be most useful. 


\begin{figure}[ht]
\sidebyside
{
\letteredcaption{b}{One caption.}}
{
\letteredcaption{c}{Two captions.}}
\end{figure}

\section{Making Tables}\inxx{Making tables}
Notice that the caption should be at the top of the table. Use
a line above the table, under the column heads, and at the
end of the table. If you use the Kluwer command, \verb+\sphline+
instead of the \LaTeX\ command \verb+\hline+, you will get
a little space added above and below the line, which will
make your table look more elegant.

This form of the tabular command makes the
table spread out to the width of the page.
This example also shows using \verb+\caption[]{}+ with the
first argument, in square brackets, used to send information
to the List of Tables. 


\begin{table}[ht]
\caption[Effects of the Two Types of Scaling Proposed by Dennard 
and Co-Workers.]%<-- this version will appear in List of Tables
{Effects of the Two Types of Scaling Proposed by \protect\inx{Dennard} 
and\newline
Co-Workers.$^{a,b}$}%<-- this version will appear on page
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcc}
\sphline
\it Parameter&\it $\kappa$ Scaling &\it $\kappa$, $\lambda$ Scaling\cr
\sphline
Dimension&$\kappa^{-1}$&$\lambda^{-1}$\cr
Voltage&$\kappa^{-1}$&$\kappa^{-1}$\cr
Currant&$\kappa^{-1}$&$\lambda/\kappa^{2}$\cr
\inx{Dopant Concentration}&$\kappa$&$\lambda^2/\kappa$\cr
\sphline
\end{tabular*}
\begin{tablenotes}
$^a$Refs.~19 and 20.

$^b\kappa, \lambda>1$.
\end{tablenotes}
\end{table}
\inxx{captions,table}

\noindent
Tables may use both the\verb+\sidebyside+ and the 
\verb+\letteredcaption+ command to position the tables
side by side and letter the captions.

\begin{table}[ht]
\sidebyside
{\letteredcaption{a}{A small table with a lettered table caption.}
\centering
\begin{tabular}{lcr}\sphline
\it $\alpha\beta\Gamma\Delta$ One&\it Two&\it Three\cr\sphline
one&two&three\cr\sphline
\end{tabular}
\label{table2a}}
{\letteredcaption{b}{A small table with a second lettered table caption.}
\centering
\begin{tabular}{lcr}\sphline
\it $\alpha\beta\Gamma\Delta$ One&\it Two&\it Three\cr\sphline
one&two&three\cr
one&two&three\cr\sphline
\end{tabular}
\label{table2b}}
\end{table}

This table shows  how you might increase vertical space between
particular lines with the use of a `strut', a vertical line with no width
so that it doesn't print, but which does have a height and/or
depth.

\begin{table}[ht]
\caption{Here is a table caption.}
\begin{center}
\begin{tabular}{||c||c||l}
\hline
  %% On the next line is an example of how to get extra vertical space in
  %% a line: Use a \vrule with width 0pt and the height or depth that you
  %% want.
\it Cell\vrule height 14pt width 0pt depth 4pt
&\it Time (sec.)&\cr
\hline
\hline
1&432.22\vrule height 12pt width0pt&\cr
%%
%% On the next line, see how to line up numbers aligned on their decimal point
2&\phantom{3}32.32&\cr
3&\phantom{33}2.32&\cr
\hline
\end{tabular}
\end{center}
\end{table}

The following table uses a continued caption, made with the command
\verb+\contcaption{}+.


\begin{table}[ht]
\contcaption{This is a continued caption.}
\begin{center}
\begin{tabular}{||c||c||l}
\hline
  %% On the next line is an example of how to get extra vertical space in
  %% a line: Use a \vrule with width 0pt and the height or depth that you
  %% want.
\it Cell\vrule height 14pt width 0pt depth 4pt
&\it \inx{Time} (sec.)&\cr
\hline
\hline
4&532.22\vrule height 12pt width0pt&\cr
%%
%% On the next line, see how to line up numbers aligned on their decimal point
5&\phantom{3}12.02&\cr
6&\phantom{33}4.44&\cr
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Figure, Table and Appendices in Landscape Mode}
These commands should be used for landscape figures, tables,
and appendices. In order for them to actually print in
landscape mode you will need to use the appropriate command
with your printer driver, a command which differs according
to which printer driver you are using. You can print
examples of these commands by uncommenting {\tt \string\end{document}}
at the end of this sample.

This is how to make a figure caption to be turned sideways on page:

{\small
\begin{verbatim}
\begin{widefigure}
\caption{This is a wide figure caption.  It is meant to be 
printed in landscape mode (sideways).  This page should be 
turned sideways when the driver program is used to translate 
the .dvi file to the file that is sent to the printer.}
\end{widefigure}
\end{verbatim}}

This is how to make a sideways table caption:

{\small
\begin{verbatim}
\begin{widetable}
\caption{This is a wide table caption.  It is meant to be 
printed in landscape mode (sideways).  This page should be 
turned sideways when the driver program is used to translate 
the .dvi file to the file that is sent to the printer.} 
\end{widetable} 
\end{verbatim}}

This is how to do an appendix printed sideways:

{\small
\begin{verbatim}
\begin{landscapemode}
\appendix{Interest Rate Liberalization Through 1988}
This is the text of the appendix.
This is the text of the appendix.
\end{landscapemode}
\end{verbatim}}


\section{Other environments}
\begin{quote}
This is a sample of extract or quotation.\inxx{quotation}%
\inxx{quotation,extract}
This is a sample of extract or quotation.
This is a sample of extract or quotation.
\end{quote}

\begin{enumerate}
\item
This is the first item in the numbered list.

\item
This is the second item in the numbered list.
This is the second item in the numbered list.
This is the second item in the numbered list.
\end{enumerate}

\begin{itemize}
\item
This is the first item in the itemized list.

\item
This is the first item in the itemized list.
This is the first item in the itemized list.
This is the first item in the itemized list.
\end{itemize}

\begin{itemize}
\item[]
This is how to get an indented paragraph without
an item marker.

\item[]
This is how to get an indented paragraph without
an item marker.
\end{itemize}


\section[Small Running Head]{Some Sample Algorithms}
When you want to demonstrate some programming code, these are
the commands to use. Lines will be preserved as you see them
on the screen, as will spaces at the beginning of the line.%
\inxx{algorithm,State transition}\inxx{algorithm}
A backslash followed with a space will indent the line. 
Blank lines will be preserved.
Math and font changes may be used. 

\begin{algorithm}
{\bf state\_transition algorithm} $\{$
\        for each neuron $j\in\{0,1,\ldots,M-1\}$
\        $\{$   
\            calculate the weighted sum $S_j$ using Eq. (6);
\            if ($S_j>t_j$)
\                    $\{$turn ON neuron; $Y_1=+1\}$   
\            else if ($S_j<t_j$)
\                    $\{$turn OFF neuron; $Y_1=-1\}$   
\            else
\                    $\{$no change in neuron state; $y_j$ remains %
unchanged;$\}$.
\        $\}$   
$\}$   
\end{algorithm}

Here is another sample algorithm:

%% \bit will produce bold italics if you are using PostScript fonts, 
%% boldface in Computer Modern.

\begin{algorithm}
{\bit Evaluate-Single-FOE} ({\bf x$_f$, I$_0$, I$_1$}):
\ {\bf I}+ := {\bf I}$_1$;
\ ($\phi,\theta$) := (0,0);
\ {\it repeat}\note{/*usually only 1 interation required*/}
\ \ (s$_{opt}${\bf E}$_\eta$) := {\bit Optimal-Shift} ({\bf I$_0$,I$^+$,I$_0$,x$_f$});
\ \ ($\phi^+$, $\theta^+$) := {\bit Equivalent-Rotation} ({\bf s}$_{opt}$);
\ \ ($\phi$, $\theta$) := ($\phi$, $\theta$) + ($\phi^+$, $\theta^+$);
\ \ {\bf I}$^+$:= {\bit Derotate-Image} ({\bf I}$_1$, $\phi$, $\theta$);
\ \ {\it until} ($\|\phi^+\|\leq\phi_{max}$ \& $\|\theta^+\|\leq\theta_{max}$);
\ {\it return} ({\bf I}$^+$, $\phi$, $\theta$, E$_\eta$).

End pseudo-code.
\end{algorithm}
\inxx{code,Pseudo}

% Notice that to produce printed `{' brackets, precede them with \string

% Notice that \begin{codebox}...\end{codebox} can be inserted within
%   codesamp, and will be positioned at the same distance from right
%   margin as text. codebox needs an argument for the width of the box,
%   as in  \begin{codebox}{2.5in} below.

This is an example of `codesamp' with a `codebox' included. Notice
that `underline' will still work even though this is basically
a verbatim environment.\inxx{code,Sample}

\begin{codesamp}
sqrdc(a, n)(a, qraux)\string{
  \underline{DARRAY float[180] a[180];}
  float qraux[180], col[180], nrmxl,t;
  DO(1=0, n)\string{
         \underline{ALIGN*(i=1, n) col[i]=a[l][i];}
         \begin{codebox}{2.3in}
         init*\string{ nrmxl=0.0;\string}
         DO*(i=l, n)\string{
           nrmxl += col[i]*col[i];\string}
         combine*\string{nrmxl;\string}
         \end{codebox}
         nmxl=sqrt(nrmxl);
         if (nrmxl != 0.00)\string{
            if (col[1]=1.0+col[1];
\end{codesamp}


\begin{glossary}
\term{GaAs}Gallium Arsinide. For similar device sizes GaAs transistors 
have three to\inxx{GaAs,Gallium Arsinide}
five times greater transconductance than those of of silicon bipolar
and MOS transistors.

\term{VLSI}Very Large Scale Integration. Since the mid-1970's 
VLSI technology has been successfully used in many areas, but its effect on
computers of all shapes and sizes has been the most dramatic. Some of the
application areas got boosts in performance while others became
feasible.
\end{glossary}


\section{Summary}
This is a \inx{summary} of this article.

\begin{acknowledgments}
The authors wish to thank Drs.~T. Misugi, M. Kobayashi, and M. Fukuta for%
\inxx{Misugi\, Dr. T.}\inxx{Kobayashi\, Dr. M.}%
\inxx{Fukuta\, Dr. M.}
their encouragement and support. Their authors also wish to thank their
colleagues...
\end{acknowledgments}

\chapappendix{}
This is a chapter appendix without a title 
meant to appear in individual chapters
of the proceedings book, not at the end of the book.

\chapappendix{This is a Chapter Appendix}
This is a chapter appendix with a title.


\begin{figure}[ht]
\caption{This is an appendix figure caption.}
\end{figure}

\begin{table}[ht]
\caption{This is an appendix table caption.}
\centering
\begin{tabular}{ccc}
\hline
one&two&three\\
\hline
C&D&E\\
\hline
\end{tabular}
\end{table}

\begin{equation}
\alpha\beta\Gamma\Delta
\end{equation}

\chapappendix{}
This is a chapter appendix without a title 
that is lettered because it is not the first
appendix.



\begin{equation}
e=mc^2
\end{equation}


\begin{chapthebibliography}{1}
\bibitem{ander}
Anderson, Terry L., and Fred S. McChesney. (n.d.). ``Raid or Trade?
An Economic Model of Indian-WhiteRelations,'' Political Economy Research
Center Working Paper 93--1.

\bibitem{lacey}
Lacey, W.K. (1968). {\it History of Socialism}. Ithaca, NY: Cornell
University Press.

\bibitem{oliva}
Oliva, Pavel. (1971). {\it Sparta and Her Social Problems.} Amsterdam: Adolf
M. Hakkert.

\bibitem{zimmern}
Zimmern, Alfred. (1961). {\it The Greek Commonwealth: Politics and Economics
in Fifth-Century Athens,}\/ 5th ed. New York: Galaxy Book, Oxford University
Press.
\end{chapthebibliography}

\articletitle{Using BibTeX for a bibliography}
\vskip48pt
\vskip1sp
\section{Sample Chapter Bibliography Using BibTeX}
If you would rather make a bibliography using Bib\TeX\ write, 
\begin{verbatim}
\bibliographystyle{apalike}
\chapbblname{chapbib}
\chapbibliography{logic}
\end{verbatim}
and substitute
the name of your .bib file for {\tt logic} below. If you don't
have apalike.bst on your system, you can get it from Kluwer at
the same .ftp site where you can find the book style files.

This will allow many Bib\TeX\ bibliographies in one book.
This example shows the chapter bibliography using
\verb+\normallatexbib+.
See the documentation, KapProc.doc, for more information.


{%\normallatexbib


\bibliographystyle{apalike}
\chapbblname{chapbib}
\chapbibliography{logic}

}


\end{document}

