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\begin{document}

\centerline{\bf Lecture~3}
\centerline{\bf Corner transfer matrices and vertex operators}
\bigskip

Consider the eight-vertex model on a large but finite lattice with
fixed boundary condition. Set the spectral parameter on the horizontal
lines to be equal to $0$, while on the vertical lines to be equal to
$u$ always except $k$ neighboring lines, where it will be equal to
$u_1,\ldots,u_k$. Let us partition the lattice into several pieces as
follows:
$$
\beginpicture
%
\setcoordinatesystem units <0.5cm,0.5cm> point at 0 0
% A matrix
\put{$A(u)$} [rB] <-4pt,4pt> at -3.25 3.25
\put{$0$} [rB] <-2pt,-2pt> at -5.72 1.0
% vertical
\rl -1.0 0.5 , -1.0 5.5
\rl -2.0 0.5 , -2.0 4.5
\rl -3.0 0.5 , -3.0 3.5
\rl -4.0 0.5 , -4.0 2.5
\rl -5.0 0.5 , -5.0 1.5
% horizontal
\rl -0.5 1.0 , -5.7 1.0 \ar -5.7 1.0 , -5.72 1.0
\rl -0.5 2.0 , -4.7 2.0 \ar -4.7 2.0 , -4.72 2.0
\rl -0.5 3.0 , -3.7 3.0 \ar -3.7 3.0 , -3.72 3.0
\rl -0.5 4.0 , -2.7 4.0 \ar -2.7 4.0 , -2.72 4.0
\rl -0.5 5.0 , -1.7 5.0 \ar -1.7 5.0 , -1.72 5.0
% B matrix
\put{$B(u)$} [rt] <-4pt,-4pt> at -3.25 -3.25
\put{$0$} [rB] <-2pt,-2pt> at -5.72 -1.0
\put{$u$} [ct] <0pt,-2pt> at -1.0 -5.72
% vertical
\rl -1.0 -0.5 , -1.0 -5.7 \ar -1.0 -5.7 , -1.0 -5.72
\rl -2.0 -0.5 , -2.0 -4.7 \ar -2.0 -4.7 , -2.0 -4.72
\rl -3.0 -0.5 , -3.0 -3.7 \ar -3.0 -3.7 , -3.0 -3.72
\rl -4.0 -0.5 , -4.0 -2.7 \ar -4.0 -2.7 , -4.0 -2.72
\rl -5.0 -0.5 , -5.0 -1.7 \ar -5.0 -1.7 , -5.0 -1.72
% horizontal
\rl -0.5 -1.0 , -5.7 -1.0 \ar -5.7 -1.0 , -5.72 -1.0
\rl -0.5 -2.0 , -4.7 -2.0 \ar -4.7 -2.0 , -4.72 -2.0
\rl -0.5 -3.0 , -3.7 -3.0 \ar -3.7 -3.0 , -3.72 -3.0
\rl -0.5 -4.0 , -2.7 -4.0 \ar -2.7 -4.0 , -2.72 -4.0
\rl -0.5 -5.0 , -1.7 -5.0 \ar -1.7 -5.0 , -1.72 -5.0
% C matrix
\put{$C(u)$} [lt] <-4pt,-4pt> at 9.25 -3.25
\put{$u$} [ct] <0pt,-2pt> at 6.0 -5.72
% vertical
\rl 6.0 -0.5 , 6.0 -5.7 \ar 6.0 -5.7 , 6.0 -5.72
\rl 7.0 -0.5 , 7.0 -4.7 \ar 7.0 -4.7 , 7.0 -4.72
\rl 8.0 -0.5 , 8.0 -3.7 \ar 8.0 -3.7 , 8.0 -3.72
\rl 9.0 -0.5 , 9.0 -2.7 \ar 9.0 -2.7 , 9.0 -2.72
\rl 10.0 -0.5 , 10.0 -1.7 \ar 10.0 -1.7 , 10.0 -1.72
% horizontal
\rl 5.5 -1.0 , 10.5 -1.0
\rl 5.5 -2.0 , 9.5 -2.0
\rl 5.5 -3.0 , 8.5 -3.0
\rl 5.5 -4.0 , 7.5 -4.0
\rl 5.5 -5.0 , 6.5 -5.0
% D matrix
\put{$D(u)$} [lB] <-4pt,4pt> at 9.25 3.25
% vertical
\rl 6.0 0.5 , 6.0 5.5
\rl 7.0 0.5 , 7.0 4.5
\rl 8.0 0.5 , 8.0 3.5
\rl 9.0 0.5 , 9.0 2.5
\rl 10.0 0.5 , 10.0 1.5
% horizontal
\rl 5.5 1.0 , 10.5 1.0
\rl 5.5 2.0 , 9.5 2.0
\rl 5.5 3.0 , 8.5 3.0
\rl 5.5 4.0 , 7.5 4.0
\rl 5.5 5.0 , 6.5 5.0
% Phi(u_1)
\put{$\Phi_{\ve_1}(u_1)$} [cB] <0pt,4pt> at 1.0 5.5
\put{$\ve_1$} [rB] <-1pt,-2pt> at 1.0 0.5
% vertical
\rl 1.0 0.5 , 1.0 5.5
% horizontal
\rl 0.5 1.0 , 1.5 1.0
\rl 0.5 2.0 , 1.5 2.0
\rl 0.5 3.0 , 1.5 3.0
\rl 0.5 4.0 , 1.5 4.0
\rl 0.5 5.0 , 1.5 5.0
%
\put{$\ldots$} [cB] at 2.5 3.0
\put{$\cdots$} [cB] <-5pt,-2pt> at 2.5 0.5
% Phi(u_k)
\put{$\Phi_{\ve_k}(u_k)$} [cB] <0pt,4pt> at 4.0 5.5
\put{$\ve_k$} [rB] <-1pt,-2pt> at 4.0 0.5
% vertical
\rl 4.0 0.5 , 4.0 5.5
% horizontal
\rl 3.5 1.0 , 4.5 1.0
\rl 3.5 2.0 , 4.5 2.0
\rl 3.5 3.0 , 4.5 3.0
\rl 3.5 4.0 , 4.5 4.0
\rl 3.5 5.0 , 4.5 5.0
% Phi^*(u_1)
\put{$\Phi^+_{\ve'_1}(u_1)$} [ct] <0pt,-14pt> at 1.0 -5.5
\put{$\ve'_1$} [rB] <-1pt,-2pt> at 1.0 -0.5
\put{$u_1$} [ct] <2pt,-2pt> at 1.0 -5.72
% vertical
\rl 1.0 -0.5 , 1.0 -5.7 \ar 1.0 -5.7 , 1.0 -5.72
% horizontal
\rl 0.5 -1.0 , 1.5 -1.0
\rl 0.5 -2.0 , 1.5 -2.0
\rl 0.5 -3.0 , 1.5 -3.0
\rl 0.5 -4.0 , 1.5 -4.0
\rl 0.5 -5.0 , 1.5 -5.0
%
\put{$\ldots$} [cB] at 2.5 -3.0
\put{$\cdots$} [cB] <-5pt,-2pt> at 2.5 -0.5
% Phi^*(u_k)
\put{$\Phi^+_{\ve'_n}(u_k)$} [ct] <0pt,-14pt> at 4.0 -5.5
\put{$\ve'_n$} [rB] <-1pt,-2pt> at 4.0 -0.5
\put{$u_k$} [ct] <2pt,-2pt> at 4.0 -5.72
% vertical
\rl 4.0 -0.5 , 4.0 -5.5 \ar 4.0 -5.7 , 4.0 -5.72
% horizontal
\rl 3.5 -1.0 , 4.5 -1.0
\rl 3.5 -2.0 , 4.5 -2.0
\rl 3.5 -3.0 , 4.5 -3.0
\rl 3.5 -4.0 , 4.5 -4.0
\rl 3.5 -5.0 , 4.5 -5.0
%
\endpicture
$$
As it is shown in the picture at each of the $k$ exceptional lines we
indeed cut the bond in the very middle and fix the variables $\ve_i$,
$\ve'_i$ at the ends.

The pieces $A(u)$, $B(u)$, $C(u)$, $D(u)$ can be considered as matrices
acting clockwise, e.g.
$$
A(u)^{\ve'_1\ve'_2\ldots}_{\ve_1\ve_2\ldots}
=\vcenter{
\beginpicture
%
\setcoordinatesystem units <0.5cm,0.5cm> point at 0 0
% A matrix
\put{$\ve_1$} [ct] <0pt,-2pt> at -1.0 0.5
\put{$\ve_2$} [ct] <0pt,-2pt> at -2.0 0.5
\put{$\cdots$} [ct] <0pt,-2pt> at -3.0 0.5
\put{$\ve'_1$} [lB] <2pt,-2pt> at -0.5 1.0
\put{$\ve'_2$} [lB] <2pt,-2pt> at -0.5 2.0
\put{$\vdots$} [lB] <2pt,-2pt> at -0.5 3.0
% vertical
\rl -1.0 0.5 , -1.0 5.5
\rl -2.0 0.5 , -2.0 4.5
\rl -3.0 0.5 , -3.0 3.5
\rl -4.0 0.5 , -4.0 2.5
\rl -5.0 0.5 , -5.0 1.5
% horizontal
\rl -0.5 1.0 , -5.7 1.0 \ar -5.7 1.0 , -5.72 1.0
\rl -0.5 2.0 , -4.7 2.0 \ar -4.7 2.0 , -4.72 2.0
\rl -0.5 3.0 , -3.7 3.0 \ar -3.7 3.0 , -3.72 3.0
\rl -0.5 4.0 , -2.7 4.0 \ar -2.7 4.0 , -2.72 4.0
\rl -0.5 5.0 , -1.7 5.0 \ar -1.7 5.0 , -1.72 5.0
%
\endpicture}
$$
The matrices $A(u)$, $B(u)$, $C(u)$, $D(u)$ are called {\it corner
transfer matrices}.

The pieces $\Phi_{\ve_i}(u_i)$ are $\Phi^+_{\ve_i}(u_i)$ act as
matrices in the left-to-right and right-to-left directions
respectively. They are called {\it half transfer matrices\/} and
(because of some properties in the infinite volume limit) {\it vertex
operators}.

We have to specify the boundary conditions at the outer boundary. We
shall fix the spins at the boundary so as if they belong to one of the
ground states described in the first lecture. We shall denote the
boundary condition by the superscript $(i)$ ($i\in\Z_2$), if
$\ve_1=(-1)^i$ in the corresponding ground state. To avoid multiple
usage of this superscript at any corner transfer matrix and vertex
operator like $A^{(i)}$, $B^{(i)}$, $C^{(i+n)}$, $D^{(i+n)}$,
$\Phi^{(i+j,i+j-1)}_{\ve_j}(u_j)$ , we shall put it at the trace signs
below.

Let $Z^{(i)}{}^{\ve_1\ldots\ve_k}_{\ve'_1\ldots\ve'_k}$ be the
partition function of the lattice with the given boundary conditions
and fixed variables at the upper and lower banks of the cut. Let
$Z^{(i)}=\sum_{\ve_1\ldots\ve_k}
Z^{(i)}{}^{\ve_1\ldots\ve_k}_{\ve_1\ldots\ve_k}$ be the partition
functions of the lattice without the cut. Now consider the ratio
$P^{(i)}{}^{\ve_1\ldots\ve_k}_{\ve'_1\ldots\ve'_k}
=Z^{(i)}{}^{\ve_1\ldots\ve_k}_{\ve'_1\ldots\ve'_n}/Z^{(i)}$. In
particular, $P^{(i)}{}^{\ve_1\ldots\ve_k}_{\ve_1\ldots\ve_k}$ is the
probability that the configuration of spins on the bonds in the middle
of the exceptional lines is $\ve_1,\ldots,\ve_k$.

These quantities are basic for calculation of local correlation
functions. For example, let us calculate the average
$\langle\ve_1\ve_2\rangle\equiv\langle\sigma^z_1\sigma^z_2\rangle$ of
the product of two neighboring spins on the lattice without any cut. It
is given by
$$
\langle\sigma^z_1\sigma^z_2\rangle^{(i)}
=P^{(i)}{}^{++}_{++}+P^{(i)}{}^{--}_{--}
-P^{(i)}{}^{+-}_{+-}-P^{(i)}{}^{-+}_{-+}.
$$
Other local correlation functions are expressed similarly.

From the partition of the lattice described above it is easy to obtain
\eq{
P^{(i)}{}^{\ve_1\ldots\ve_k}_{\ve'_1\ldots\ve'_n}
={1\over Z^{(i)}}\Tr^{(i)}(\Phi^+_{\ve'_1}(u_1)\ldots\Phi^+_{\ve'_n}(u_k)
C(u)D(u)\Phi_{\ve_k}(u_k)\ldots\Phi_{\ve_1}(u_1)A(u)B(u)).
\label{PTrace}
}
Surely, we have not yet approached the exact solution to the problem.
Nevertheless, in the large volume limit the objects defined above get
remarkable properties.

First of all, not all of these objects are independent. From the
crossing symmetry it is easy to find that
\eq{
C(u)=QA(u)Q,
\qquad
B(u)=QD(u)Q=A(1-u)Q,
\qquad
\Phi^+_\ve(u)=Q\Phi_\ve(u)Q,
\label{ABCDCrossing}
}
where $Q=\sigma^x\otimes\sigma^x\otimes\ldots$ is the operator that
flips all spins.

Baxter observed~\cite{BaxterCTM} that in the large volume limit
\eq{
A(u)=F(u)e^{-\epsilon uH}
\label{ALimit}
}
with some scalar (not operator) function $F(u)$ and some constant
operator $H$ with the discrete spectrum $\{0,1,2,\ldots\}$.

Let us sketch Baxter's argumentation. Consider the product
$A(u)B(u-v)$. On the infinite lattice this product, considered as a
vector, must be an eigenvector corresponding to the largest eigenvalue
of the transfer matrix of an inhomogeneous model. Hence, the product
$A(u)A(1+v-u)$ as a function of $u$ is a constant operator times a
scalar function. As $A(0)=1$, we have the equation
\eq{
A(u)A(u')=g(u,u')A(u+u').
\label{Adiff}
}
Therefore
$$
A(u)A'(0)=g_{u'}(u,0)A(u)+g(u,0)A'(u)
$$
or
$$
{A'(u)\over A(u)}=f_1(u)A'(0)+f_2(u).
$$
Solving this differential equation we obtain
$$
A(u)=F_1(u)e^{F_2(u)A'(0)},
$$
with some functions $F_1(u)$, $F_2(u)$, such that
$$
F_1(0)=1,
\qquad
F_2(0)=0,
\qquad
F'_2(0)=1.
$$
Substituting this solution back into the difference equation
(\ref{Adiff}), we obtain that $F_2(u)=u$. We obtain
\eq{
A(u)=F_1(u)e^{uA'(0)}.
\label{Asolution}
}
From the definition of the model we know that the Boltzmann weight
without the function $\rho(u)$ are doubly periodic. They are
unchanged after the substitutions $u\to u+2r$ and $u\to
u+2\i\pi/\epsilon$. The function $\rho(u)$ written out in the first
lecture do not respect the first periodicity and respects the second.
This is very important. The function $\rho(u)$ can be obtained directly
from the Bethe ansatz, i.e.\ from first principles. Moreover, it can be
concluded from the Bethe ansatz solution that all physical quantities
respect this second periodicity. Therefore, impose this periodicity on
the solution~(\ref{Asolution}). We immediately obtain that the spectrum
of $A'(u)$ is equidistant with the separation $\epsilon$. This
`proves' (on the physical level of rigorousness) the
equation~(\ref{ALimit}). From now on we shall omit the factor $F(u)$ and
write
\eq{
A(u)=e^{-\epsilon uH}=z^{H/2}.
\label{Arenormalized}
}

Now, on the physical level of rigorousness, it is easy to obtain the
following two commutation relations~\cite{JMbook}:
\eq{
\Aligned{
{}\span
\Phi_\ve(u)Q=Q\Phi_{-\ve}(u),
\\
\span
\Phi_\ve(v)A(u)=A(u)\Phi_\ve(v-u),
\\
\span
\Phi_{\ve_1}(u_1)\Phi_{\ve_2}(u_2)
=\sum_{\ve'_1\ve'_2}R(u_1-u_2)^{\ve'_1\ve'_2}_{\ve_1\ve_2}
\Phi_{\ve'_2}(u_2)\Phi_{\ve'_1}(u_1).
}\label{AQPhiCommut}
}
The first equation is trivial. The second equation can be proven as
follows. Take the product $\Phi_\ve(v)A(u)$ and, using the Yang--Baxter
equation, push the line corresponding to $\Phi_\ve$ to the left:
$$
\vcenter{
\beginpicture
%
\setcoordinatesystem units <0.5cm,0.5cm> point at 0 0
%
\put{$v$} [tc] <0pt,-2pt> at 0.1 -0.5
%
\put{$u$} [tc] <0pt,-2pt> at -1.0 -0.5
\put{$u$} [tc] <0pt,-2pt> at -2.0 -0.5
\put{$u$} [tc] <0pt,-2pt> at -3.0 -0.5
\put{$u$} [tc] <0pt,-2pt> at -4.0 -0.5
\put{$u$} [tc] <0pt,-2pt> at -5.0 -0.5
% vertical
\rl 0.1 -0.5 , 0.1 5.5 \ar 0.1 -0.5 , 0.1 -0.52
%
\rl -1.0 -0.5 , -1.0 5.5 \ar -1.0 -0.5 , -1.0 -0.52
\rl -2.0 -0.5 , -2.0 4.5 \ar -2.0 -0.5 , -2.0 -0.52
\rl -3.0 -0.5 , -3.0 3.5 \ar -3.0 -0.5 , -3.0 -0.52
\rl -4.0 -0.5 , -4.0 2.5 \ar -4.0 -0.5 , -4.0 -0.52
\rl -5.0 -0.5 , -5.0 1.5 \ar -5.0 -0.5 , -5.0 -0.52
% horizontal
\rl 0.6 1.0 , -5.7 1.0 \ar -5.7 1.0 , -5.72 1.0
\rl 0.6 2.0 , -4.7 2.0 \ar -4.7 2.0 , -4.72 2.0
\rl 0.6 3.0 , -3.7 3.0 \ar -3.7 3.0 , -3.72 3.0
\rl 0.6 4.0 , -2.7 4.0 \ar -2.7 4.0 , -2.72 4.0
\rl 0.6 5.0 , -1.7 5.0 \ar -1.7 5.0 , -1.72 5.0
%
\endpicture
}
\qquad\longrightarrow\qquad
\vcenter{
\beginpicture
%
\setcoordinatesystem units <0.5cm,0.5cm> point at 0 0
%
\put{$v$} [tc] <0pt,-2pt> at 0.1 -0.5
%
\put{$u$} [tc] <0pt,-2pt> at -1.0 -0.5
\put{$u$} [tc] <0pt,-2pt> at -2.0 -0.5
\put{$u$} [tc] <0pt,-2pt> at -3.0 -0.5
\put{$u$} [tc] <0pt,-2pt> at -4.0 -0.5
\put{$u$} [tc] <0pt,-2pt> at -5.0 -0.5
% vertical
\ln 0.1 5.5 , -5.5 0.2 \rl -5.5 0.2 , 0.1 0.2
\rl 0.1 0.2 , 0.1 -0.5 \ar 0.1 -0.5 , 0.1 -0.52
%
\rl -1.0 -0.5 , -1.0 5.5 \ar -1.0 -0.5 , -1.0 -0.52
\rl -2.0 -0.5 , -2.0 4.5 \ar -2.0 -0.5 , -2.0 -0.52
\rl -3.0 -0.5 , -3.0 3.5 \ar -3.0 -0.5 , -3.0 -0.52
\rl -4.0 -0.5 , -4.0 2.5 \ar -4.0 -0.5 , -4.0 -0.52
\rl -5.0 -0.5 , -5.0 1.5 \ar -5.0 -0.5 , -5.0 -0.52
% horizontal
\rl 0.6 1.0 , -5.7 1.0 \ar -5.7 1.0 , -5.72 1.0
\rl 0.6 2.0 , -4.7 2.0 \ar -4.7 2.0 , -4.72 2.0
\rl 0.6 3.0 , -3.7 3.0 \ar -3.7 3.0 , -3.72 3.0
\rl 0.6 4.0 , -2.7 4.0 \ar -2.7 4.0 , -2.72 4.0
\rl 0.6 5.0 , -1.7 5.0 \ar -1.7 5.0 , -1.72 5.0
%
\endpicture
}
$$
Any physicist knows that the boundary does not affect essentially the
bulk. So let us forget completely about the skew line on the right
picture. Its only effect is the boundary condition. The lower
horizontal line is just the operator $\Phi_\ve(u-v)$.

The derivation of the last line is similar. Consider the product in the
r.h.s.\ and push the $R$ matrix upside:
$$
\vcenter{
\beginpicture
%
\setcoordinatesystem units <0.5cm,0.5cm> point at 0 0
%
% vertical
\rl 0.0 5.5 , 0.0 0.0 \ln 0.0 0.0 , 1.0 -1.0
\rl 1.0 -1.0 , 1.0 -2.0 \ar 1.0 -2.0 , 1.0 -2.02
\rl 1.0 5.5 , 1.0 0.0 \ln 1.0 0.0 , 0.0 -1.0
\rl 0.0 -1.0 , 0.0 -2.0 \ar 0.0 -2.0 , 0.0 -2.02
% horizontal
\rl 2.0 5.0 , -1.0 5.0 \ar -1.0 5.0 , -1.02 5.0
\rl 2.0 4.0 , -1.0 4.0 \ar -1.0 4.0 , -1.02 4.0
\rl 2.0 3.0 , -1.0 3.0 \ar -1.0 3.0 , -1.02 3.0
\rl 2.0 2.0 , -1.0 2.0 \ar -1.0 2.0 , -1.02 2.0
\rl 2.0 1.0 , -1.0 1.0 \ar -1.0 1.0 , -1.02 1.0
%
\put{$u_1$} [tc] <0pt,-2pt> at 1.0 -2.02
\put{$u_2$} [tc] <0pt,-2pt> at 0.0 -2.02
\put{$\ve_1$} [lB] <2pt,-2pt> at 1.0 -1.5
\put{$\ve_2$} [rB] <-2pt,-2pt> at 0.0 -1.5
\put{$\Phi(u_1)$} [rb] <0pt,2pt> at 0.0 5.5
\put{$\Phi(u_2)$} [lb] <0pt,2pt> at 1.0 5.5
\put{$R(u_1-u_2)$} [rB] <-6pt,-2pt> at 0.5 -0.5
%
\endpicture
}
\qquad\longrightarrow\qquad
\vcenter{
\beginpicture
%
\setcoordinatesystem units <0.5cm,0.5cm> point at 0 0
%
% vertical
\rl 0.0 5.5 , 0.0 5.0 \ln 0.0 5.0 , 1.0 4.0
\rl 1.0 4.0 , 1.0 -2.0 \ar 1.0 -2.0 , 1.0 -2.02
\rl 1.0 5.5 , 1.0 5.0 \ln 1.0 5.0 , 0.0 4.0
\rl 0.0 4.0 , 0.0 -2.0 \ar 0.0 -2.0 , 0.0 -2.02
% horizontal
\rl 2.0 3.0 , -1.0 3.0 \ar -1.0 3.0 , -1.02 3.0
\rl 2.0 2.0 , -1.0 2.0 \ar -1.0 2.0 , -1.02 2.0
\rl 2.0 1.0 , -1.0 1.0 \ar -1.0 1.0 , -1.02 1.0
\rl 2.0 0.0 , -1.0 0.0 \ar -1.0 0.0 , -1.02 0.0
\rl 2.0 -1.0 , -1.0 -1.0 \ar -1.0 -1.0 , -1.02 -1.0
%
\put{$u_1$} [tc] <0pt,-2pt> at 1.0 -2.02
\put{$u_2$} [tc] <0pt,-2pt> at 0.0 -2.02
\put{$\ve_1$} [lB] <2pt,-2pt> at 1.0 -1.5
\put{$\ve_2$} [rB] <-2pt,-2pt> at 0.0 -1.5
\put{$\Phi(u_2)$} [rb] <-2pt,-2pt> at -1.02 3.0
\put{$\Phi(u_1)$} [lb] <2pt,-2pt> at 2.0 3.0
\put{$R(u_1-u_2)$} [rB] <-6pt,-2pt> at 0.5 4.5
%
\endpicture
}
$$
Now the $R$ matrix `at the infinity' can be erased and we obtain the
third equation.

From these equations we derive that
$$
\Phi^+_\ve(u_j)C(u)D(u)=
Q\Phi_{-\ve}(u_j)e^{-2\epsilon H}=Qe^{-2\epsilon H}\Phi_{-\ve}(u_j-1)
=C(u)D(u)\Phi_{-\ve}(u_j-1).
$$
Let us introduce the notation
\eq{
\Phi^*_\ve(u)=\Phi_{-\ve}(u-1).
\label{PhiStarDef}
}
Then
\eq{
\Phi^+_\ve(u_j)C(u)D(u)=C(u)D(u)\Phi^*_\ve(u_j).
\label{Phi+Phi*}
}
It means that we can move the product $C(u)D(u)$ to the left
simultaneously replacing $\Phi^+_\ve(u_j)$ by $\Phi^*_\ve(u_j)$.
From the fact that $\sum_\ve\Phi_\ve(u)\otimes(\Phi^+_\ve(u))^t$ is just
the transfer matrix on the infinite lattice with the largest eigenvalue
1 we conclude that
\eq{
\sum_\ve\Phi^*_\ve(u)\Phi_\ve(u)=1,
\qquad
\Phi_\ve(u)\Phi^*_{\ve'}(u)=\delta_{\ve\ve'}.
\label{Phi*Properties}
}

Now consider the product of the product of the corner transfer matrices.
It is easy to find from (\ref{Arenormalized}) that
\eq{
A(u)B(u)C(u)D(u)=e^{-4\epsilon H}=x^{2H}.
\label{ABCDProd}
}
We can also specify what we mean under $\Tr^{(i)}$ in the infinite
volume limit. Consider the consequences of spin variables
$\ve(1),\ve(2),\ldots$ that stabilize to $\ve(n)=(-)^{i+n}$. The space
of such {\it paths\/} will be denoted by $\cH^{(i)}$. Then
$\Tr^{(i)}=\Tr_{\cH^{(i)}}$.


Substituting (\ref{Phi+Phi*}) and (\ref{ABCDProd}) to (\ref{PTrace}) we
obtain
\eq{
P^{(i)}{}^{\ve_1\ldots\ve_k}_{\ve'_1\ldots\ve'_n}
={1\over\chi^{(i)}}\Tr_{\cH^{(i)}}
(\Phi^*_{\ve'_1}(u_1)\ldots\Phi^*_{\ve'_n}(u_k)
\Phi_{\ve_k}(u_k)\ldots\Phi_{\ve_1}(u_1)x^{2H}).
\label{PTraceFinal}
}
with
\eq{
\chi^{(i)}=\Tr_{\cH^{(i)}}x^{2H}.
\label{chiDef}
}
It was shown \cite{DFJMN}, that
$$
\Tr_{\cH^{(i)}}q^H={1\over(q;q^2)_\infty}.
$$
This result was proven in the limit $x\to0$, $r\to\infty$, but, since
degeneracy cannot change continuously, it holds in the whole $\rm AF_1$
phase.

Generally, we can consider the {\it trace functions}
\eq{
F^{(i)}_{\ve_1\ldots\ve_k}(u_1,\ldots,u_k)
={1\over\chi^{(i)}}\Tr_{\cH^{(i)}}
(\Phi_{\ve_k}(u_k)\ldots\Phi_{\ve_1}(u_1)x^{2H}).
\label{TraceFuncDef}
}
Evidently,
$$
P^{(i)}{}^{\ve_1\ldots\ve_k}_{\ve'_1\ldots\ve'_n}
=F^{(i)}_{\ve_1,\ldots,\ve_k,-\ve'_n,\ldots,-\ve'_1}
(u_1,\ldots,u_k,u_k-1,\ldots,u_1-1).
$$
The functions $F^{(i)}(u_1,\ldots,u_k)$ satisfy a number of difference
equations, that follow from the properties of the corner transfer
matrices and vertex operators:
\Align{
\span
F^{(i)}_{\ve_1\ldots\ve_k}(u_1+v,\ldots,u_k+v)
=F^{(i)}_{\ve_1\ldots\ve_k}(u_1,\ldots,u_k),
\label{FHomogen}
\\
\span
F^{(i)}_{\ve_1\ve_2\ldots\ve_k}(u_1+2\i\pi/\epsilon,u_2,\ldots,u_k)
=F^{(i)}_{\ve_1\ve_2\ldots\ve_k}(u_1,u_2,\ldots,u_k),
\label{FPeriod}
\\
\span
F^{(i)}_{\ve_1\ve_2\ldots\ve_k}(u_1,u_2,\ldots,u_k)
=F^{(i+1)}_{\ve_2\ldots\ve_k\ve_1}(u_2,\ldots,u_k,u_1-2),
\label{FCyclic}
\\
\span
\sum_\ve F^{(i)}_{\ve_1,\ldots,\ve_k,\ve,-\ve}
(u_1,\ldots,u_k,u,u-1)
=F^{(i)}_{\ve_1\ldots\ve_k}(u_1,\ldots,u_k),
\label{FNorm}
\\
\span
F^{(i)}_{\ldots\ve_j\ve_{j+1}\ldots}(\ldots,u_j,u_{j+1},\ldots)
=\sum_{\ve'_j\ve'_{j+1}}
R(u_{j+1}-u_j)^{\ve'_j\ve'_{j+1}}_{\ve_j\ve_{j+1}}
F^{(i)}_{\ldots\ve'_{j+1}\ve'_j\ldots}(\ldots,u_{j+1},u_j,\ldots).
\label{FCommut}
}
In principle, it is possible to find the functions $F$ and
probabilities~$P$ by solving these equations under some analyticity
conditions. In practice, the case $n=2$ in $F$ only admits a direct
solution. In more general case we need some additional ideas to solve
the equations.

Consider now the SOS model. The necessary partition of the lattice looks
like:
$$
\beginpicture
%
\setcoordinatesystem units <0.5cm,0.5cm> point at 0 0
% A matrix
\put{$A_{mn}(u)$} [rb] <-4pt,4pt> at -3.25 3.25
\put{$0$} [rB] <-2pt,-2pt> at -6.72 1.0
% vertical
\rl -0.5 5.5 , -0.5 0.5
\rl -1.5 5.5 , -1.5 0.5
\rl -2.5 4.5 , -2.5 0.5
\rl -3.5 3.5 , -3.5 0.5
\rl -4.5 2.5 , -4.5 0.5
\rl -5.5 1.5 , -5.5 0.5
% horizontal
\rl -5.5 0.5 , -0.5 0.5
\rl -5.5 1.5 , -0.5 1.5
\rl -4.5 2.5 , -0.5 2.5
\rl -3.5 3.5 , -0.5 3.5
\rl -2.5 4.5 , -0.5 4.5
\rl -1.5 5.5 , -0.5 5.5
\setdashes
\rl -5.0 1.0 , -6.7 1.0
\setsolid
\ar -6.7 1.0 , -6.72 1.0
% B matrix
\put{$B_{mn}(u)$} [rt] <-4pt,-4pt> at -3.25 -4.25
\put{$0$} [rB] <-2pt,-2pt> at -6.72 -2.0
\put{$u$} [tc] <0pt,-2pt> at -1.0 -7.72
% vertical
\rl -0.5 -6.5 , -0.5 -1.5
\rl -1.5 -6.5 , -1.5 -1.5
\rl -2.5 -5.5 , -2.5 -1.5
\rl -3.5 -4.5 , -3.5 -1.5
\rl -4.5 -3.5 , -4.5 -1.5
\rl -5.5 -2.5 , -5.5 -1.5
% horizontal
\rl -5.5 -1.5 , -0.5 -1.5
\rl -5.5 -2.5 , -0.5 -2.5
\rl -4.5 -3.5 , -0.5 -3.5
\rl -3.5 -4.5 , -0.5 -4.5
\rl -2.5 -5.5 , -0.5 -5.5
\rl -1.5 -6.5 , -0.5 -6.5
\setdashes
\rl -5.0 -2.0 , -6.7 -2.0
\rl -1.0 -6.0 , -1.0 -7.7
\setsolid
\ar -6.7 -2.0 , -6.72 -2.0
\ar -1.0 -7.7 , -1.0 -7.72
% C matrix
\put{$C_{mn'}(u)$} [lt] <4pt,-4pt> at 8.25 -4.25
\put{$u$} [tc] <0pt,-2pt> at 6.0 -7.72
% vertical
\rl 5.5 -6.5 , 5.5 -1.5
\rl 6.5 -6.5 , 6.5 -1.5
\rl 7.5 -5.5 , 7.5 -1.5
\rl 8.5 -4.5 , 8.5 -1.5
\rl 9.5 -3.5 , 9.5 -1.5
\rl 10.5 -2.5 , 10.5 -1.5
% horizontal
\rl 10.5 -1.5 , 5.5 -1.5
\rl 10.5 -2.5 , 5.5 -2.5
\rl 9.5 -3.5 , 5.5 -3.5
\rl 8.5 -4.5 , 5.5 -4.5
\rl 7.5 -5.5 , 5.5 -5.5
\rl 6.5 -6.5 , 5.5 -6.5
\setdashes
\rl -5.0 -2.0 , -6.7 -2.0
\rl 6.0 -6.0 , 6.0 -7.7
\setsolid
\ar -6.7 -2.0 , -6.72 -2.0
\ar 6.0 -7.7 , 6.0 -7.72
% D matrix
\put{$D_{mn'}(u)$} [lb] <4pt,4pt> at 8.25 3.25
% vertical
\rl 5.5 5.5 , 5.5 0.5
\rl 6.5 5.5 , 6.5 0.5
\rl 7.5 4.5 , 7.5 0.5
\rl 8.5 3.5 , 8.5 0.5
\rl 9.5 2.5 , 9.5 0.5
\rl 10.5 1.5 , 10.5 0.5
% horizontal
\rl 10.5 0.5 , 5.5 0.5
\rl 10.5 1.5 , 5.5 1.5
\rl 9.5 2.5 , 5.5 2.5
\rl 8.5 3.5 , 5.5 3.5
\rl 7.5 4.5 , 5.5 4.5
\rl 6.5 5.5 , 5.5 5.5
% Phi(u_1)
\put{$\Phi(u_1)$} [cB] <0pt,4pt> at 1.0 5.5
\put{$n$} [ct] <0pt,-2pt> at 0.0 0.5
\put{$n_1$} [ct] <0pt,-2pt> at 1.5 0.5
% vertical
\rl 1.5 0.5 , 1.5 5.5
\rl 0.5 0.5 , 0.5 5.5
% horizontal
\rl 0.5 0.5 , 1.5 0.5
\rl 0.5 1.5 , 1.5 1.5
\rl 0.5 2.5 , 1.5 2.5
\rl 0.5 3.5 , 1.5 3.5
\rl 0.5 4.5 , 1.5 4.5
\rl 0.5 5.5 , 1.5 5.5
%
\put{$\ldots$} [cB] at 2.5 3.0
% Phi(u_k)
\put{$\Phi(u_k)$} [cB] <0pt,4pt> at 4.0 5.5
\put{$n_{k-1}$} [ct] <0pt,-2pt> at 3.5 0.5
\put{$n'$} [ct] <0pt,-2pt> at 5.0 0.5
% vertical
\rl 4.5 0.5 , 4.5 5.5
\rl 3.5 0.5 , 3.5 5.5
% horizontal
\rl 3.5 0.5 , 4.5 0.5
\rl 3.5 1.5 , 4.5 1.5
\rl 3.5 2.5 , 4.5 2.5
\rl 3.5 3.5 , 4.5 3.5
\rl 3.5 4.5 , 4.5 4.5
\rl 3.5 5.5 , 4.5 5.5
% Phi^+(u_1)
\put{$\Phi^+(u_1)$} [ct] <0pt,-2pt> at 1.0 -6.5
\put{$n$} [cb] <0pt,2pt> at 0.0 -1.5
\put{$n'_1$} [cb] <0pt,2pt> at 1.5 -1.5
\put{$u_1$} [ct] <0pt,-2pt> at 1.0 -7.72
% vertical
\rl 1.5 -1.5 , 1.5 -6.5
\rl 0.5 -1.5 , 0.5 -6.5
\setdashes
\ln 1.0 -6.0 , 1.0 -6.5
\ar 1.0 -7.7 , 1.0 -7.72
\setsolid
% horizontal
\rl 0.5 -1.5 , 1.5 -1.5
\rl 0.5 -2.5 , 1.5 -2.5
\rl 0.5 -3.5 , 1.5 -3.5
\rl 0.5 -4.5 , 1.5 -4.5
\rl 0.5 -5.5 , 1.5 -5.5
\rl 0.5 -6.5 , 1.5 -6.5
%
\put{$\ldots$} [cB] at 2.5 -4.0
% Phi(u_k)
\put{$\Phi^+(u_k)$} [ct] <0pt,-2pt> at 4.0 -6.5
\put{$n'_{k-1}$} [cb] <0pt,2pt> at 3.5 -1.5
\put{$n'$} [cb] <0pt,2pt> at 5.0 -1.5
\put{$u_k$} [ct] <0pt,-2pt> at 4.0 -7.72
% vertical
\rl 4.5 -1.5 , 4.5 -6.5
\rl 3.5 -1.5 , 3.5 -6.5
\setdashes
\ln 4.0 -6.0 , 4.0 -6.5
\ar 4.0 -7.7 , 4.0 -7.72
\setsolid
% horizontal
\rl 3.5 -1.5 , 4.5 -1.5
\rl 3.5 -2.5 , 4.5 -2.5
\rl 3.5 -3.5 , 4.5 -3.5
\rl 3.5 -4.5 , 4.5 -4.5
\rl 3.5 -5.5 , 4.5 -5.5
\rl 3.5 -6.5 , 4.5 -6.5
%
\endpicture
$$
Here we introduce the corner transfer matrices
$A_{mn}(u),\ldots,D_{mn}(u)$, which depend on the `central site'
variable $n$ and the boundary condition $(m,m+1)$ or $(m+1,m)$
depending on the parity of $n-m$, and the vertex operators
$\Phi(u_j)^{n_j}_{n_{j-1}}$ and $\Phi^+(u_j)^{n'_{j-1}}_{n'_j}$, which
depend on the variables at their feet.

In the same way as for the eight-vertex model one can write the
quantities
\Multline{
P_m\Bigl(n\Matrix{n_1\ldots n_{k-1}\\n'_1\ldots n'_{k-1}}n'\Bigr)
={1\over Z_m}\Tr(
\Phi^+(u_1)^{n_1}_{n'_1}\ldots\Phi^+(u_k)^{n'_{k-1}}_{n'}
C_{mn}(u)D_{mn}(u)
\\
\times
\Phi(u_k)^{n'}_{n_{k-1}}\ldots\Phi(u_1)^{n_1}_{n}
A_{mn}(u)B_{mn}(u)).
\label{SOSPTrace}
}
In particular, for $n'_i=n_i$ ($i=1,\ldots,k-1$) they are the
multipoint {\it local height probabilities\/}, which describe the
probabilities of configurations along a finite line on the lattice.

In the infinite volume limit we have up to a scalar factor
\eq{
A_{mn}(u)=e^{-2\epsilon uH_{mn}}.
\label{SOSALimit}
}
The product of the corner transfer matrices is given by
\eq{
A_{mn}(u)B_{mn}(u)C_{mn}(u)D_{mn}(u)=[n]x^{4H_{mn}}.
\label{SOSABCDProd}
}
The additional factor $2$ before the corner Hamiltonian in comparison
with the eight-vertex model is related to the (quasi)periodicity of all
quantities with the period $\i\pi/\epsilon$ instead of
$2\pi\i/\epsilon$, while the factor $[n]$ in the product is related to
the similar factors in the crossing property.

Another important property is
\eq{
\Phi^+(u)^{n'}_nC_{mn}(v)D_{mn}(v)
=C_{mn'}(v)D_{mn'}(v)\Phi^*(u)^{n'}_n,
\label{SOSPhi+Phi*}
}
where
\eq{
\Phi^*(u)^{n'}_n=[n]\Phi(u-1)^{n'}_n,
\qquad
\sum_{n'}\Phi^*(u)^n_{n'}\Phi(u)^{n'}_n=1,
\qquad
\Phi(u)^{n'}_{n''}\Phi^*(u)^{n''}_n=\delta^{n'}_n.
\label{SOSPhi*Def}
}
The basic commutation relations look like
\Align{
\span
\Phi(u)^{n'}_nx^{2vH_{mn}}=x^{2vH_{mn'}}\Phi(u-v)^{n'}_n,
\label{SOSPhiACommut}
\\
\span
\Phi(u_1)^{n'}_s\Phi(u_2)^s_n
=\sum_{s'}\W(s,n';s',n|u_1-u_2)
\Phi(u_2)^{n'}_{s'}\Phi(u_1)^{s'}_n.
\label{SOSPhiPhiCommut}
}

In the infinite volume limit define the trace functions
\eq{
F^m_{nn_1\ldots n_k}(u_1,\ldots,u_k)
={[n]\over\chi_m}\Tr_{\cH_{mn}}(\Phi(u_k)^n_{n_k}\ldots
\Phi(u_2)^{n_2}_{n_1}\Phi(u_1)^{n_1}_nx^{4H_{mn}})
\label{SOSTraceFunction}
}
with
\eq{
\chi_m=\sum_n[n]\chi_{mn},
\qquad
\chi_{mn}=\Tr_{\cH_{mn}}x^{4H_{mn}}.
\label{SOSchiDef}
}
Here $\cH_{mn}$ is the space of paths $n(0)=n,n(1),n(2),\ldots$ that
stabilize to the sequence $\ldots,m,m+1,m,m+1,\ldots$. There is a
remarkable result by Andrews, Baxter and Forrester~\cite{ABF} based on
the $x\to0$ limit (the {\it low temperature\/} limit) that
$$
\Tr_{\cH_{mn}}q^{H_{mn}}={q^{(rm-(r-1)n)^2/4r(r-1)}\over(q;q)_\infty}.
$$
Notice an important property of $\chi_{mn}$:
\eq{
\sum_{n\in2\Z+m+i}[n]\chi_{mn}=[m]'\chi^{(i)},
\qquad
[u]'=[u]|_{r\to r-1}.
\label{ChiSum}
}
From the properties (\ref{SOSABCDProd}) and (\ref{SOSPhi+Phi*}) we
obtain
$$
P_m\Bigl(n\Matrix{n_1\ldots n_{k-1}\\n'_1\ldots n'_{k-1}}n'\Bigr)
=F^m_{nn_1\ldots n_{k-1}n'n'_{k-1}\ldots n'_1}
(u_1,\ldots,u_k,u_k-1,\ldots,u_1-1)
[n']\prod^{k-1}_{j=1}[n'_k].
$$

Using the commutation relations (\ref{SOSPhiACommut}),
(\ref{SOSPhiPhiCommut}) we obtain
\Align{
\span
F^m_{nn_1\ldots n_{k-1}}(u_1+v,\ldots,u_k+v)
=F^m_{nn_1\ldots n_{k-1}}(u_1,\ldots,u_k),
\label{SOSFHomogen}
\\
\span
F^m_{nn_1n_2\ldots n_{k-1}}(u_1+\i\pi/\epsilon,u_2,\ldots,u_k)
=-e^{\i\pi(n_1^2-n^2)/2r}F^m_{nn_1n_2\ldots n_k}(u_1,u_2,\ldots,u_k),
\label{SOSFPeriod}
\\
\span
F^m_{nn_1n_2\ldots n_k}(u_1,u_2,\ldots,u_k)
={[n]\over[n_1]}F^m_{n_1n_2\ldots n_kn}(u_2,\ldots,u_k,u_1),
\label{SOSFCyclic}
\\
\span
\sum_{n'}[n']F^m_{nn_1\ldots n_{k-1}nn'}(u_1,\ldots,u_k,u,u-1)
=F^m_{nn_1\ldots n_{k_1}}(u_1,\ldots,u_k),
\label{SOSFNorm}
\\
\span
F^m_{\ldots n_{k-1}n_kn_{k+1}\ldots}(\ldots,u_k,u_{k+1},\ldots)
=\sum_{n'_k}\W(n_k,n_{k+1};n'_k,n_{k-1}|u_{k+1}-u_k)
F^m_{\ldots n_{k-1}n'_kn_{k+1}\ldots}(\ldots,u_{k+1},u_k,\ldots).
\label{SOSFCommut}
}
The problem of solving these equations in the SOS model will be
discussed in the next lecture. The respective problem for the
eight-vertex model is more difficult and will be the topic of the last
lecture.

\begin{thebibliography}{9}
\itemsep=0pt\parskip=0pt
\bibitem{BaxterCTM}
R.~J.~Baxter, {\it J.~Stat.\ Phys.}\ {\bf15} (1976) 485;
{\it J.~Stat.\ Phys.}\ {\bf17} (1977) 1
\bibitem{JMbook}
M.~Jimbo, T.~Miwa, {\it Algebraic analysis of solvable lattice models},
CBMS Regional Conference Series in Mathematics {\bf85} AMS (1994)
\bibitem{DFJMN}
B.~Davies, O.~Foda, M.~Jimbo, T.~Miwa, and A.~Nakayashiki,
{\it J.~Phys.}\ {\bf A26} (1993) 2199
\bibitem{ABF}
G.~E.~Andrews, R.~J.~Baxter, and P.~J.~Forrester,
{\it J.\ Stat.\ Phys.}\ {\bf 35} (1984) 193
\end{thebibliography}









\end{document}
