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\begin{document}

\centerline{\bf Lecture~2}
\centerline{\bf SOS model and vertex-face correspondence}
\bigskip

To understand better the origin of the SOS model let us sketch the
Bethe ansatz for the six-vertex model, where $d=0$. In this case we can
introduce the operator $S^z$ of `total spin', which counts the signs
along a column:
$$
S^z(v_{\ve_1}\otimes\cdots\otimes v_{\ve_N})
={1\over2}\sum^N_{i=1}\ve_i(v_{\ve_1}\otimes\cdots\otimes v_{\ve_N}).
$$
Due to the ice condition this operator commutes with the transfer matrix
$$
[T(u),S^z]=0.
$$
This is a trivial fact: the number of `minuses' is conserved.

So we can easily establish at least two eigenvectors ({\it
pseudovacuums\/})
$$
|\Omega_\pm\rangle=v_\pm\otimes\cdots\otimes v_\pm
$$
with the eigenvalue $a^N+b^N$. But this is generally (everywhere except
in the ferroelectric regions ${\rm F}_i$) NOT the largest one.
How to find the other eigenvectors? Let us start from
$|\Omega_+\rangle$ and flip spins one by one. Any state with the
eigenvalue of $S^z$ being $N/2-n$ will be called a state of $n$
pseudoparticles. Let $\sigma^-_k=(\sigma^x-i\sigma^y)/2$ be the
operator that turns the $k$th spin down.

Consider the state of one pseudoparticle. From the translational
invariance we conclude, that it has the form
$$
|p\rangle=\sum^N_{k=1}e^{\i pk}\sigma^-_k|\Omega_+\rangle.
$$
From cyclic boundary condition we conclude that
$$
e^{ipN}=1,
$$
so that we have $N$ states with $p_j={2\pi\over N}j$, $j=0,\ldots,N-1$.
You can easily find the corresponding eigenvalues. There are larger ones
than $a^N+b^N$, but they are also do not contain the largest one.

Consider the state of $2$ pseudoparticles. Substitute the following
ansatz:
$$
|p_1,p_2\rangle=\sum_{k_1<k_2}(A_{12}e^{ip_1k_1+ip_2k_2}
+A_{21}e^{\i p_2k_1+\i p_1k_2})
\sigma^-_{k_1}\sigma^-_{k_2}|\Omega_+\rangle.
$$
Apply the operator $T$ or, simpler, $H_1$. A miracle! This is an
eigenvector if
$$
{A_{12}\over A_{21}}=z(p_1,p_2)
$$
with some given function $z(p_1,p_2)$. The cyclic boundary condition
imposes the restrictions
$$
e^{\i p_1N}=z(p_1,p_2),
\qquad
e^{\i p_2N}=z(p_2,p_1).
$$
In the general case of $n$ pseudoparticles the same miracle takes
place. The wave function can be made of plane waves. The cyclic
boundary conditions impose the {\it Bethe equations}
$$
e^{\i p_jN}=\prod^n_{j'(\ne j)}z(p_j,p_{j'}),
\qquad
j=1,\ldots,n.
$$
It is generally impossible to solve these equations analytically. But in
the limit $N\to\infty$, $n/N=\const$ they are reduced to an integral
equation. The case $n=N/2$, corresponding to the largest eigenvalue,
admits an analytic solution. This is how the six-vertex model is
solved.

What is wrong with the general eight-vertex model? The obstacle is that
$$
[S^z,T]\ne0\quad\text{for $d\ne0$.}
$$
It destroys the whole picture of pseudoparticles. There is
a nice construction of the $Q$ operator proposed by Baxter, that makes
it possible to obtain the Bethe equations without any reference to the
Bethe ansatz. Nevertheless, there is a question: is it possible to
relate this model to another one that admits the whole construction of
Bethe ansatz? Is it possible to construct something similar to the
six-vertex model, but involving elliptic functions? The answer is YES.

Consider again the square lattice on the plane, but associate the
variables to the vertices of the lattice and the Boltzmann weights to
the plaquet or faces. Namely, associate to each vertex a variable
$n\in\Z+\delta$, where the real shift $\delta$ is introduced for
convenience. The partition function will be independent of this shift.
Associate to each face of the lattice a weight:
$$
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\put{$e^{-E(n_1,n_2,n_3,n_4)/T}
=\W(n_1,n_2;n_3,n_4|u-v)=$} [rB] at 3.7 2.4
\rl 5.0 2.0 , 6.0 2.0
\rl 6.0 2.0 , 6.0 3.0
\rl 6.0 3.0 , 5.0 3.0
\rl 5.0 3.0 , 5.0 2.0
\setdashes <2.5pt>
\rl 5.5 3.3 , 5.5 1.5
\rl 6.3 2.5 , 4.5 2.5
\setsolid
\ar 5.5 1.51 , 5.5 1.5
\ar 4.51 2.5 , 4.5 2.5
\put{$n_1$} [tr] at 4.9 1.9
\put{$n_2$} [tl] at 6.1 1.9
\put{$n_3$} [bl] at 6.1 3.1
\put{$n_4$} [br] at 4.9 3.1
\put{$u$} [t]  at 5.5 1.4
\put{$v$} [rB] at 4.4 2.5
%
\endpicture
$$
The dashed lines, first, denote the orientation and, second, carry the
spectral parameters. The configuration sum is taken over all $n$s at all
vertices such that
\eq{
|n_i-n_j|=1
\qquad\text{\bf(admissibility condition)}
\label{AdmissibilityCondition}
}
on the neighboring vertices.

What does the admissibility condition mean? Consider the dual (dashed)
lattice. Define on each edge of this lattice a variable $\ve=+1$ if the
variable $n_i=n_j+1$, if $i$ denotes the vertex on the left or upper
end of the edge, while $j$ denotes the vertex on the right or lower end
of the edge:
$$
\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\rl 5.0 2.0 , 6.0 2.0
\rl 6.0 2.0 , 6.0 3.0
\rl 6.0 3.0 , 5.0 3.0
\rl 5.0 3.0 , 5.0 2.0
\setdashes <2.5pt>
\rl 5.5 3.3 , 5.5 1.5
\rl 6.3 2.5 , 4.5 2.5
\setsolid
\ar 5.5 1.51 , 5.5 1.5
\ar 4.51 2.5 , 4.5 2.5
\put{$n+\ve_1$} [tr] at 4.9 1.9
\put{$n$} [tl] at 6.1 1.9
\put{$n+\ve'_2$} [bl] at 6.1 3.1
\put{$n+\ve_1+\ve_2$} [br] at 4.9 3.1
%
\endpicture
}
\quad
\longrightarrow
\quad
\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\setdashes <2.5pt>
\rl 5.5 3.3 , 5.5 1.5
\rl 6.3 2.5 , 4.5 2.5
\setsolid
\ar 5.5 1.51 , 5.5 1.5
\ar 4.51 2.5 , 4.5 2.5
\put{$\ve_1$} [Br] <-2pt,-2pt> at 5.5 2.1
\put{$\ve_2$} [bc] <0pt,2pt> at 5.0 2.5
\put{$\ve'_1$} [Br] <-2pt,-2pt> at 5.5 3.2
\put{$\ve'_2$} [bc] <0pt,2pt> at 6.0 2.5
\put{\strut} [Br] <-2pt,-2pt> at 5.5 3.3
%
\endpicture
}
\qquad
(n+\ve_1+\ve_2=n+\ve'_1+\ve'_2)
$$
In these notations the variables $\ve$ satisfy the ice condition by
definition. But the weight $W$ at each vertex of the dual lattice
depend not only on the variables $\ve_1,\ve_2,\ve'_1,\ve'_2$, but also
on the value of $n$ at e.g.\ the right lower corner of the face, which
is (up to $\delta$) the sum of all $\ve$s on any path along the initial
(solid) lattice from some fixed point at the lattice to this right lower
corner of the face.

The Boltzmann weights, analogous to $a$, $b$ and $c$ of the six-vertex
model, are given by
\eq{
\Aligned{
a^\pm_n(u)=\W(n\pm1,n;n\pm1,n\pm2|u)
&=R_0(u),
\cr
b^\pm_n(u)=\W(n\pm1,n;n\mp1,n|u)
&=R_0(u){[n\mp1][u]\over [n][1-u]}
\cr
c^\pm_n(u)=\W(n\pm1,n;n\pm1,n|u)
&=R_0(u){[n\pm u][1]\over [n][1-u]},
}\label{Wmatrix}
}
with an arbitrary function $R_0(u)$ and
$$
\Aligned{{}
[u]_i
&=\sqrt{\pi\over\epsilon r}\,e^{\quarter\epsilon r}\,
\theta_i\!\left({u\over r};{\i\pi\over\epsilon r}\right),
\\
[u]
&=[u]_1=x^{u^2/r-u}(z;p)_\infty(pz^{-1};p)_\infty(p;p)_\infty.
}
$$
The weights $W$ satisfy the Yang--Baxter equation of the form
\Multline{
\sum_n\W(n,n'_3;n_2,n'_1|u_1-u_2)
\W(n_2,n_1;n'_3,n|u_1-u_3)
\W(n_3,n_2;n,n'_1|u_2-u_3)
\\
=\sum_n\W(n,n_1;n'_3,n'_2|u_2-u_3)
\W(n_3,n;n'_2,n'_1|u_1-u_3)
\W(n_2,n_1;n,n_3|u_1-u_2).
\label{WYBEu}
}
Graphically it looks like:
$$
\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\rl -1.0 -0.5 , -1.0 0.5
\rl 0.0 -1.0 , 0.0 0.0
\rl 1.0 -0.5 , 1.0 0.5
\ln -1.0 -0.5 , 0.0 -1.0
\ln 0.0 -1.0 , 1.0 -0.5
\ln -1.0 0.5 , 0.0 1.0
\ln 0.0 1.0 , 1.0 0.5
\ln 1.0 0.5 , 0.0 0.0
\ln 0.0 0.0 , -1.0 0.5
\setdashes
\rl -0.5 1.25 , -0.5 0.75
\ln -0.5 0.75 , 0.5 0.25
\rl 0.5 0.25 , 0.5 -1.25 %\ar 0.5 -1.23 , 0.5 -1.25
\rl 0.5 1.25 , 0.5 0.75
\ln 0.5 0.75 , -0.5 0.25
\rl -0.5 0.25 , -0.5 -1.25 %\ar -0.5 -1.23 , -0.5 -1.25
\rl 1.5 0.0 , 1.0 0.0
\ln 1.0 0.0 , 0.0 -0.5
\ln 0.0 -0.5 , -1.0 0.0
\rl -1.0 0.0 , -1.5 0.0 %\ar -1.48 0.0 , -1.5 0.0
\setsolid
\ar 0.5 -1.23 , 0.5 -1.25
\ar -0.5 -1.23 , -0.5 -1.25
\ar -1.48 0.0 , -1.5 0.0
\put{$n_1$} [lt] <2pt,-2pt> at 1.0 -0.5
\put{$n_2$} [ct] <0pt,-2pt> at 0.0 -1.0
\put{$n_3$} [rt] <-2pt,-2pt> at -1.0 -0.5
\put{$n'_1$} [rb] <-2pt,2pt> at -1.0 0.5
\put{$n'_2$} [cb] <0pt,2pt> at 0.0 1.0
\put{$n'_3$} [lb] <2pt,2pt> at 1.0 0.5
\put{$u_1$} [tc] <0pt,-2pt> at 0.5 -1.25
\put{$u_2$} [tc] <0pt,-2pt> at -0.5 -1.25
\put{$u_3$} [Br] <-2pt,-2pt> at -1.5 0.0
%
\endpicture
}
\quad=\quad
\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\rl -1.0 -0.5 , -1.0 0.5
\rl 0.0 1.0 , 0.0 0.0
\rl 1.0 -0.5 , 1.0 0.5
\ln -1.0 -0.5 , 0.0 -1.0
\ln 0.0 -1.0 , 1.0 -0.5
\ln 1.0 -0.5 , 0.0 0.0
\ln 0.0 0.0 , -1.0 -0.5
\ln -1.0 0.5 , 0.0 1.0
\ln 0.0 1.0 , 1.0 0.5
\setdashes
\rl -0.5 1.25 , -0.5 -0.25
\ln -0.5 -0.25 , 0.5 -0.75
\rl 0.5 -0.75 , 0.5 -1.25 %\ar 0.5 -1.23 , 0.5 -1.25
\rl 0.5 1.25 , 0.5 -0.25
\ln 0.5 -0.25 , -0.5 -0.75
\rl -0.5 -0.75 , -0.5 -1.25 %\ar -0.5 -1.23 , -0.5 -1.25
\rl 1.5 0.0 , 1.0 0.0
\ln 1.0 0.0 , 0.0 0.5
\ln 0.0 0.5 , -1.0 0.0
\rl -1.0 0.0 , -1.5 0.0 %\ar -1.48 0.0 , -1.5 0.0
\setsolid
\ar 0.5 -1.23 , 0.5 -1.25
\ar -0.5 -1.23 , -0.5 -1.25
\ar -1.48 0.0 , -1.5 0.0
\put{$n_1$} [lt] <2pt,-2pt> at 1.0 -0.5
\put{$n_2$} [ct] <0pt,-2pt> at 0.0 -1.0
\put{$n_3$} [rt] <-2pt,-2pt> at -1.0 -0.5
\put{$n'_1$} [rb] <-2pt,2pt> at -1.0 0.5
\put{$n'_2$} [cb] <0pt,2pt> at 0.0 1.0
\put{$n'_3$} [lb] <2pt,2pt> at 1.0 0.5
\put{$u_1$} [tc] <0pt,-2pt> at 0.5 -1.25
\put{$u_2$} [tc] <0pt,-2pt> at -0.5 -1.25
\put{$u_3$} [Br] <-2pt,-2pt> at -1.5 0.0
%
\endpicture
}
$$
The dashed lines here play the role of solid lines in the Yang--Baxter
equation for the eight-vertex model, while the solid lines here simply
form the lattice dual to the dashed one.

If the function $R_0(u)$ satisfy the relations
$$
R_0(u)R_0(-u)=1,
\qquad
R_0(u)[u]=R_0(1-u)[1-u],
$$
the weights satisfy the relations
\Align{
\span
\sum_n\W(n_1,n_2;n,n_4|u)\W(n,n_2;n_3,n_4|-u)
=\delta_{n_1,n_3},
&&\qquad
\text{\bf(Unitarity)},
\label{WUnitarity}
\\
\span
[n_3]^{-1}\W(n_1,n_2;n_3,n_4|u)
=[n_4]^{-1}\W(n_2,n_3;n_4,n_1|1-u)
&&\qquad
\text{\bf(Crossing symmetry)}.
\label{WCrossing}
}
The solution
\eq{
R_0(u)\equiv R_0(u;\epsilon,r)=z^{(r-1)/2r}{g(z^{-1})\over g(z)}
\label{R0Def}
}
makes the partition function per site equal to~$1$.

It is easy to define the $L$ operator
\eq{
L(u)^{n'_1\ldots n'_{N+1}}_{n_1\ldots n_{N+1}}
=\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\rl 0.0 0.0 , 0.0 1.1
\setdashes
\rl 0.0 1.1 , 0.0 1.8
\setsolid
\rl 0.0 1.8 , 0.0 2.5
\rl 0.5 0.0 , 0.5 1.1
\setdashes
\rl 0.5 1.1 , 0.5 1.8
\setsolid
\rl 0.5 1.8 , 0.5 2.5
\rl 0.0 0.0 , 0.5 0.0
\rl 0.0 0.5 , 0.5 0.5
\rl 0.0 1.0 , 0.5 1.0
\rl 0.0 2.0 , 0.5 2.0
\rl 0.0 2.5 , 0.5 2.5
\put{$n_1$} [rB] <-2pt,-2pt> at 0.0 0.0
\put{$n_2$} [rB] <-2pt,-2pt> at 0.0 0.5
\put{$n_3$} [rB] <-2pt,-2pt> at 0.0 1.0
\put{$n_N$} [rB] <-2pt,-2pt> at 0.0 2.0
\put{$n_{N+1}$} [rB] <-2pt,-2pt> at 0.0 2.5
\put{$n'_1$} [lB] <2pt,-2pt> at 0.5 0.0
\put{$n'_2$} [lB] <2pt,-2pt> at 0.5 0.5
\put{$n'_3$} [lB] <2pt,-2pt> at 0.5 1.0
\put{$n'_N$} [lB] <2pt,-2pt> at 0.5 2.0
\put{$n'_{N+1}$} [lB] <2pt,-2pt> at 0.5 2.5
%
\endpicture}
=\W(n_N,n'_N;n'_{N+1},n_{N+1}|u)\ldots
\W(n_2,n'_2;n'_3,n_3|u)\W(n_1,n'_1;n'_2,n_2|u)
\label{WLDef}
}
and the transfer matrix
\eq{
T(u)^{n'_1\ldots n'_N}_{n_1\ldots n_N}
=L(u)^{n'_1\ldots n'_Nn'_1}_{n_1\ldots n_Nn_1}.
\label{WTDef}
}
The transfer matrices form a commuting family,
$$
T(u_1)T(u_2)=T(u_2)T(u_1),
$$
and $T(0)$ is again the shift operator.

Now we formulate Baxter's fundamental statement about the relation
between two models~\cite{BaxterII73}. There exist functions
$t_\ve(u)^{n'}_n$ such that
\eq{
\sum_{\ve'_1\ve'_2}R(u-v)_{\ve_1\ve_2}^{\ve'_1\ve'_2}
t_{\ve'_2}(v-u_0)^n_{s'}t_{\ve'_1}(u-u_0)^{s'}_{n'}
=\sum_{s\in\Z}t_{\ve_1}(u-u_0)^n_st_{\ve_2}(v-u_0)^s_{n'}
\W(s,n;s',n'|u-v)
\label{VertexFace}
}
for arbitrary $u_0$. This relation is referred to as the {\it
vertex--face correspondence}. Explicitly, these {\it intertwining\/}
functions have the form
\eq{
\Aligned{
t_+(u)^{n'}_n
&=(-1)^{(n-\delta)(n'-n-1)/2}e^{\i\pi/4}f(u)
\theta_3\!\left({(n'-n)u+n'\over2r};\i{\pi\over2\epsilon r}\right),
\\
t_-(u)^{n'}_n
&=-(-1)^{(n-\delta)(n'-n+1)/2}e^{-\i\pi/4}f(u)
\theta_4\!\left({(n'-n)u+n'\over2r};\i{\pi\over2\epsilon r}\right).
}\label{Intertwining}
}
Here $f(u)$ is an arbitrary function and $\delta$ is the shift discussed
above.

To understand better the fundamental identity (\ref{VertexFace}), let us
represent it graphically. Introduce the graphical representative of the
intertwining functions:
$$
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\put{$t_\ve(u-u_0)^{n'}_n=$} [rB] at 2.8 4.2
\trl 4.0 4.5 , 5.0 4.5
\ln 4.4 4.5 , 4.5 4.4
\ln 4.5 4.4 , 4.6 4.5
\rl 4.5 4.4 , 4.5 3.3 \ar 4.5 3.31 , 4.5 3.3
\setdashes <2.5pt>
\rl 3.8 4.2 , 5.2 4.2
\setsolid
\ar 3.81 4.2 , 3.8 4.2
\put{$n'$} [b] at 5.0 4.6
\put{$n$} [b] at 4.0 4.6
\put{$\ve$} [lB] at 4.6 3.8
\put{$u$} [t] at 4.5 3.2
\put{$u\smash{{}_0}$} [r] at 3.7 4.2
%
\endpicture
$$
With this notation the vertex--face correspondence looks like ($u_0$
line is not depicted)
$$
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\trl 1.0 2.5 , 2.0 2.5
\ln 1.4 2.5 , 1.5 2.4 \ln 1.5 2.4 , 1.6 2.5
\trl 2.0 2.5 , 2.0 1.5
\ln 2.0 2.1 , 1.9 2.0 \ln 1.9 2.0 , 2.0 1.9
\rl 1.5 2.4 , 1.5 1.1 \ar 1.5 1.11 , 1.5 1.1
\rl 1.9 2.0 , 0.6 2.0 \ar 0.61 2.0 , 0.6 2.0
\put{$n$}  [lt] at 2.1 1.4
\put{$s'$} [lB] at 2.1 2.6
\put{$n'$} [rB] at 0.9 2.6
\put{$\ve_1$} [lB] at 1.6 1.4
\put{$\ve_2$} [lt] at 1.0 1.9
\put{$u$} [t]  at 1.5 1.0
\put{$v$} [rB] at 0.5 1.95
\put{$=$} [B] at 2.75 1.95
\trl 4.0 2.5 , 4.0 1.5
\ln 4.0 2.1 , 3.9 2.0 \ln 3.9 2.0 , 4.0 1.9
\trl 4.0 1.5 , 5.0 1.5
\ln 4.4 1.5 , 4.5 1.4 \ln 4.5 1.4 , 4.6 1.5
\rl 4.0 2.5 , 5.0 2.5
\rl 5.0 2.5 , 5.0 1.5
\rl 3.9 2.0 , 3.4 2.0 \ar 3.41 2.0 , 3.4 2.0
\rl 4.5 1.4 , 4.5 0.9 \ar 4.5 0.91 , 4.5 0.9
\setdashes <2.5pt>
\rl 5.3 2.0 , 4.0 2.0
\rl 4.5 2.8 , 4.5 1.5
\setsolid
\put{$n$}  [lt] at 5.1 1.4
\put{$s'$} [lB] at 5.1 2.6
\put{$n'$} [rB] at 3.9 2.6
\put{$\ve_1$} [lB] at 4.65 1.1
\put{$\ve_2$} [lt] at 3.6 1.85
\put{$u$} [t]  at 4.5 0.8
\put{$v$} [rB] at 3.3 1.95
%
\endpicture
$$
Note that this relation looks like the Yang--Baxter equation of mixed
vertex--face type!

This means that if we take a square finite SOS lattice with open
boundaries and attach intertwining functions to their left and lower
boundaries summing over necessary boundary variables $n$, we can push
the intertwining functions up and right using the vertex--face
correspondence and obtain a square lattice of the eight-vertex model
with the intertwining functions attached to the right and upper
boundaries. In physics we usually expect that the contribution of
boundaries to the partition function is neglectable in a large system.
It means that the large volume limit of the partition functions per
site of the eight-vertex model and of the SOS model coincide.

Moreover, it can be rigorously derived that the spectra of eigenvalues
of the transfer matrices of the eight-vertex and SOS models with the
cyclic boundary condition coincide. First, introduce the $L$-type
operator
\eq{
\lambda(u_0)^{n_1\ldots n_Nn_{N+1}}_{\ve_1\ldots\ve_N}
=\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\trl 0.0 -0.25 , 0.0 0.75
\setdashes
\trl 0.0 0.75 , 0.0 1.75
\setsolid
\trl 0.0 1.75 , 0.0 2.25
\ln 0.0 -0.1 , -0.1 0.0 \ln -0.1 0.0 , 0.0 0.1
\rl -0.1 0.0 , -0.7 0.0 \ar -0.68 0.0 , -0.7 0.0
\ln 0.0 0.4 , -0.1 0.5 \ln -0.1 0.5 , 0.0 0.6
\rl -0.1 0.5 , -0.7 0.5 \ar -0.68 0.5 , -0.7 0.5
\ln 0.0 1.9 , -0.1 2.0 \ln -0.1 2.0 , 0.0 2.1
\rl -0.1 2.0 , -0.7 2.0 \ar -0.68 2.0 , -0.7 2.0
\put{$\ve_1$} [rB] <-2pt,-2pt> at -0.7 0.0
\put{$\ve_2$} [rB] <-2pt,-2pt> at -0.7 0.5
\put{$\ve_N$} [rB] <-2pt,-2pt> at -0.7 2.0
\put{$n_1$} [lB] <2pt,-2pt> at 0.0 -0.25
\put{$n_2$} [lB] <2pt,-2pt> at 0.0 0.25
\put{$n_3$} [lB] <2pt,-2pt> at 0.0 0.75
\put{$n_N$} [lB] <2pt,-2pt> at 0.0 1.75
\put{$n_{N+1}$} [lB] <2pt,-2pt> at 0.0 2.25
%
\endpicture
}
=t_{\ve_N}(-u_0)^{n_{N-1}}_{n_N}\ldots\,
t_{\ve_2}(-u_0)^{n_2}_{n_3}t_{\ve_1}(-u_0)^{n_1}_{n_2}
}
and the transfer matrix type operator
$$
\tau(u_0)^{n_1\ldots n_N}_{\ve_1\ldots\ve_N}
=\lambda(u_0)^{n_1\ldots n_Nn_1}_{\ve_1\ldots\ve_N}.
$$
Then
$$
\sum_{n_2\ldots n_{N-1}}L(u)^{n'_1\ldots n'_{N+1}}_{n_1\ldots n_{N+1}}
\lambda(u_0)^{n_1\ldots n_{N+1}}t_1(u-u_0)^{n_1}_{n'_1}
=\sum_n\lambda(u_0)^{n'_1\ldots n'_{N+1}}
t_1(u-u_0)^{n'_{N+1}}_nL_1(u)
$$
or, graphically,
$$
\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\trl 0.0 0.0 , 0.0 1.1
\setdashes
\trl 0.0 1.1 , 0.0 1.8
\setsolid
\trl 0.0 1.8 , 0.0 2.5
\rl 0.5 0.0 , 0.5 1.1
\setdashes
\rl 0.5 1.1 , 0.5 1.8
\setsolid
\rl 0.5 1.8 , 0.5 2.5
\trl 0.0 0.0 , 0.5 0.0
\rl 0.0 0.5 , 0.5 0.5
\rl 0.0 1.0 , 0.5 1.0
\rl 0.0 2.0 , 0.5 2.0
\rl 0.0 2.5 , 0.5 2.5
\ln 0.0 0.15 , -0.1 0.25 \ln -0.1 0.25 , 0.0 0.35
\rl -0.1 0.25 , -0.68 0.25 \ar -0.68 0.25 , -0.7 0.25
\ln 0.0 0.65 , -0.1 0.75 \ln -0.1 0.75 , 0.0 0.85
\rl -0.1 0.75 , -0.68 0.75 \ar -0.68 0.75 , -0.7 0.75
\ln 0.0 2.15 , -0.1 2.25 \ln -0.1 2.25 , 0.0 2.35
\rl -0.1 2.25 , -0.68 2.25 \ar -0.68 2.25 , -0.7 2.25
\ln 0.15 0.0 , 0.25 -0.1 \ln 0.25 -0.1 , 0.35 0.0
\rl 0.25 -0.1 , 0.25 -0.48 \ar 0.25 -0.48 , 0.25 -0.5
\put{$n_{N+1}$} [cb] <-2pt,2pt> at 0.0 2.5
\put{$n'_1$} [lB] <2pt,-2pt> at 0.5 0.0
\put{$n'_2$} [lB] <2pt,-2pt> at 0.5 0.5
\put{$n'_3$} [lB] <2pt,-2pt> at 0.5 1.0
\put{$n'_N$} [lB] <2pt,-2pt> at 0.5 2.0
\put{$n'_{N+1}$} [lB] <2pt,-2pt> at 0.5 2.5
\put{$\ve$} [tc] <0pt,-2pt> at 0.25 -0.5
\put{$\ve_1$} [rB] <-0.2pt,-0.2pt> at -0.7 0.25
\put{$\ve_2$} [rB] <-0.2pt,-0.2pt> at -0.7 0.75
\put{$\ve_N$} [rB] <-0.2pt,-0.2pt> at -0.7 2.25
%
\endpicture
}
\quad=\quad
\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
\trl 0.5 0.0 , 0.5 1.1
\setdashes
\trl 0.5 1.1 , 0.5 1.8
\setsolid
\trl 0.5 1.8 , 0.5 2.5
\trl 0.0 2.5 , 0.5 2.5
\ln 0.5 0.15 , 0.4 0.25 \ln 0.4 0.25 , 0.5 0.35
\rl 0.4 0.25 , -0.68 0.25 \ar -0.68 0.25 , -0.7 0.25
\ln 0.5 0.65 , 0.4 0.75 \ln 0.4 0.75 , 0.5 0.85
\rl 0.4 0.75 , -0.68 0.75 \ar -0.68 0.75 , -0.7 0.75
\ln 0.5 2.15 , 0.4 2.25 \ln 0.4 2.25 , 0.5 2.35
\rl 0.4 2.25 , -0.68 2.25 \ar -0.68 2.25 , -0.7 2.25
\ln 0.15 2.5 , 0.25 2.4 \ln 0.25 2.4 , 0.35 2.5
\rl 0.25 2.4 , 0.25 -0.48 \ar 0.25 -0.48 , 0.25 -0.5
\put{$n_{N+1}$} [cb] <-2pt,2pt> at 0.0 2.5
\put{$n'_1$} [lB] <2pt,-2pt> at 0.5 0.0
\put{$n'_2$} [lB] <2pt,-2pt> at 0.5 0.5
\put{$n'_3$} [lB] <2pt,-2pt> at 0.5 1.0
\put{$n'_N$} [lB] <2pt,-2pt> at 0.5 2.0
\put{$n'_{N+1}$} [lB] <2pt,-2pt> at 0.5 2.5
\put{$\ve$} [tc] <0pt,-2pt> at 0.25 -0.5
\put{$\ve_1$} [rB] <-0.2pt,-0.2pt> at -0.7 0.25
\put{$\ve_2$} [rB] <-0.2pt,-0.2pt> at -0.7 0.75
\put{$\ve_N$} [rB] <-0.2pt,-0.2pt> at -0.7 2.25
%
\endpicture
}
$$
Introduce now the object `inverse' to the intertwining functions:
\eq{
\sum_\ve t_\ve(u)^n_{n'}t^*_\ve(u)^{n''}_n=\delta_{n'n''}
\quad\text{or}\quad
\sum_{n'}t^*_{\ve'}(u)^{n'}_nt_\ve(u)^n_{n'}=\delta_{\ve\ve'}.
\label{t^+Def}
}
or, graphically,
$$
t^*_\ve(u-u_0)^{n'}_n
=\quad\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at 0 0
%
\trl 4.0 1.5 , 5.0 1.5
\ln 4.4 1.5 , 4.5 1.6
\ln 4.5 1.6 , 4.6 1.5
\rl 4.5 1.6 , 4.5 2.4
\setdashes <2.5pt>
\rl 4.5 0.9 , 4.5 1.5
\rl 3.8 1.8 , 5.2 1.8
\setsolid
\ar 4.5 0.91 , 4.5 0.9
\ar 5.2 1.8 , 5.21 1.8
\put{$n\smash{{}'}$} <-2pt,-3pt> [rt] at 4.0 1.5
\put{$n$} <2pt,-3pt> [lt] at 5.0 1.5
\put{$\ve$} [lB] at 4.6 2.0
\put{$u$} [t] at 4.5 0.8
\put{$u\smash{{}_0}$} [lB] <2pt,-2pt> [r] at 5.21 1.8
%
\endpicture
}\quad,
\qquad\qquad
\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at -7 0
\trl 1.0 4.5 , 2.0 4.5
\ln 1.4 4.5 , 1.5 4.4 \ln 1.5 4.4 , 1.6 4.5
\trl 1.0 3.5 , 2.0 3.5
\ln 1.4 3.5 , 1.5 3.6 \ln 1.5 3.6 , 1.6 3.5
\rl 1.5 4.4 , 1.5 3.6
\setdots <2.5pt>
\rl 2.0 4.5 , 2.0 3.5
\setsolid
\put{$n'$} [rB] at 0.9 4.6
\put{$n\smash{{}''}$} [rt] at 0.9 3.4
\put{$n$} [lt] at 2.1 3.4
\put{$n$} [lB] at 2.1 4.6
\put{$t$}   [B] at 1.5 4.6
\put{$t^*$} [t] at 1.5 3.4
%
\endpicture
}\quad=\delta_{n'n''},
\qquad\qquad
\sum_{n'}\left(
\vcenter{
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at -7 0
\trl 1.0 4.0 , 2.0 4.0
\ln 1.4 4.0 , 1.5 3.9 \ln 1.5 3.9 , 1.6 4.0
\ln 1.4 4.0 , 1.5 4.1 \ln 1.5 4.1 , 1.6 4.0
\rl 1.5 4.5 , 1.5 4.1 \rl 1.5 3.9 , 1.5 3.32 \ar 1.5 3.32 , 1.5 3.3
\put{$n'$} [rB] <-2pt,-2pt> at 1.0 4.0
\put{$n$} [lB] <2pt,-2pt> at 2.0 4.0
\put{$\ve$} [lB] <2pt,-2pt> at 1.5 3.3
\put{$\ve'$} [lB] <2pt,-2pt> at 1.5 4.5
%
\endpicture
}\right)
\quad=\delta_{\ve\ve'}.
$$
Attaching these $t^*$ functions to the upper boundary and imposing the
cyclic boundary condition we obtain
\eq{
\tau(u_0)T_{\rm8v}(u)=T_{\rm SOS}(u)\tau(u_0),
\label{VertexFaceTau}
}
where $T_{\rm8v}$ and $T_{\rm SOS}(u)$ are transfer matrices of the
eight-vertex and SOS models respectively. Note that this equation has
been obtained in the full analogy to the derivation of commutativity of
transfer matrices.

Similarly, one can introduce the matrix
$$
\tau^*(u)^{\ve_1\ldots\ve_N}_{n_1\ldots n_N}
=t^*_{\ve_N}(-u_0)^{n_1}_{n_N}\ldots t^*_{\ve_1}(-u_0)^{n_2}_{n_1}
$$
with the relation
\eq{
T_{\rm8v}(u)\tau^*(u_0)=\tau^*(u_0)T_{\rm SOS}(u),
\label{VertexFaceTau*}
}


Let $|\Lambda\rangle_{\rm SOS}$ be an eigenvector of $T_{\rm SOS}(u)$
with the eigenvalue function $\Lambda(u)$. Then
$$
T_{\rm8v}(u)\tau^*(u_0)|\Lambda\rangle_{\rm SOS}
=\tau^*(u_0)T_{\rm SOS}(u)|\Lambda\rangle_{\rm SOS}
=\Lambda(u)\tau^*(u_0)|\Lambda\rangle_{\rm SOS}.
$$
It means that $|\Lambda\rangle_{\rm8v}=\tau^*(u_0)|\Lambda\rangle_{\rm
SOS}$ is an eigenvector of the operator $T_{\rm8v}(u)$ with the same
eigenvalue function $\Lambda(u)$. It proves that the spectra of both
models coincide.

To conclude, let us say something about the ground states in this
theory. We shall consider the SOS model in the so called regime~III
region:
$$
\epsilon>0,
\qquad
r\ge1,
\qquad
0<u<1.
$$
In this region the ground states (the states of maximal weight) are
numerated by $m\in\Z+\delta$ and $m'=m\pm1$, such that $(k-1)r<m,m'<kr$
for some integer $k$. The ground state $(m,m')$ looks like
$$
\beginpicture
%
\setcoordinatesystem units <1cm,1cm> point at -7 0
%
\rl 1.0 1.0 , 4.0 1.0
\rl 1.0 2.5 , 4.0 2.5
\rl 1.0 4.0 , 4.0 4.0
\rl 1.0 1.0 , 1.0 4.0
\rl 2.5 1.0 , 2.5 4.0
\rl 4.0 1.0 , 4.0 4.0
\put{$m'$} [lb] at 1.1 4.1
\put{$m'$} [lb] at 4.1 4.1
\put{$m'$} [lb] at 2.6 2.6
\put{$m'$} [lb] at 1.1 1.1
\put{$m'$} [lb] at 4.1 1.1
\put{$m$} [lb] at 2.6 4.1
\put{$m$} [lb] at 1.1 2.6
\put{$m$} [lb] at 4.1 2.6
\put{$m$} [lb] at 2.6 1.1
%
\endpicture
$$

The conclusion is the following. There is a highly nonlocal
transformation that relates the eight-vertex model to another model,
the solid-on-solid one, which can be treated by means of the Bethe ansatz
approach. Though this relation is not a direct one-to-one correspondence
between configurations, it is nevertheless a `detailed' correspondence
that makes it possible to express any expectation value of the
eight-vertex model to an expectation value defined in terms of the SOS
model. We discuss this point in the Lecture~4.


\begin{thebibliography}{9}
\itemsep=0pt\parskip=0pt
\bibitem{BaxterII73}
R.~J.~Baxter, Annals Phys.\ {\bf76} (1973) 25--47
\end{thebibliography}








\end{document}
