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Conformal Field Theory, Lattice models, relativistic field theory,
solvable models, quantum Knizhnik-Zamolodchikov equation, spin chains,
form factors
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\begin{document}
\begin{center}
{\bf Connecting lattice and relativistic models
via conformal field theory.}
\end{center}
\phantom{a}
\vspace{1.5cm}
\centerline{H. E. Boos
\footnote{on leave of absence from the Institute for High Energy Physics,
Protvino, 142284, Russia}}
\centerline{\it Max-Planck Institut f{\"u}r Mathematik}
\centerline{\it Vivatsgasse 7, 53111 Bonn, Germany}
\phantom{a}
\vspace{0.5cm}
\phantom{a}
\centerline{V. E. Korepin }
\centerline{\it C.N.~Yang Institute for Theoretical Physics}
\centerline{\it State University of New York at Stony Brook}
\centerline{\it Stony Brook, NY 11794--3840, USA}
\phantom{a}
\vspace{0.5cm}
\phantom{a}
\centerline{F.A. Smirnov
\footnote{Membre du CNRS}
}
\centerline{\it LPTHE, Tour 16, 1-er {\'e}tage, 4, pl. Jussieu}
\centerline{\it 75252, Paris Cedex 05, France}
\vspace{1cm}
\vskip2em
\begin{abstract}
\noindent
We consider the quantum group invariant XXZ-model.
In infrared limit it describes Conformal Field Theory (CFT)
with modified energy-momentum tensor. The correlation functions
are related to solutions of level -4 of qKZ equations. We describe
these solutions relating them to level 0 solutions. We further consider
general matrix elements (form factors) containing local operators
and asymptotic states. We explain that the formulae for solutions
of qKZ equations suggest a decomposition of these matrix
elements with respect to states of corresponding CFT.
\end{abstract}
\newpage
\section
{Quantum group invariant XXZ-model.}
Let us recall some well known facts concerning XXZ-model and
its continuous limit. Usually XXZ-model is considered as thermodynamic
limit of finite spin chain.
Consider the space $\(\mathbb{C}^2\)^{\otimes N}$. The finite spin chain in question
is described by the Hamiltonian:
\begin{align}
H_{XXZ}=\sum\limits _{k=1}^N(\si ^1_k\si _{k+1}^1+\si ^2_k\si _{k+1}^2+
\Delta\si ^3_k\si _{k+1}^3)\label{hxxz}
\end{align}
where the periodic boundary conditions are implied: $\si _{N+1}=\si _1$.
We consider the critical case $|\Delta |<1$ and parametrize it as follows:
$$\Delta =\cos \pi \nu$$
It is well-known that in the infrared limit the model
describes Conformal Field Theory (CFT) with $c=1$ and coupling constant
equal to $\nu$. The correlation functions in the thermodynamic limit
were found by Jimbo and Miwa \cite{jm}.
It is equally matter of common knowledge that the model is closely
related to the R-matrix:
\begin{align}
R(\b, \nu)=
%\frac { R_0(\b)} {\b+\pi i}
\left(
\begin{array}{cccc}
a(\b)&0&0&0\\
0&b(\b)&c(\b)&0\\
0&c(\b)&b(\b)&0\\
0&0&0&a(\b)
\end{array}
\right)
\label{R-m}
\end{align}
where
\begin{align}
&a(\b)= R_0(\b),\quad b(\b)= -R_0(\b)\frac {\sinh \nu \b}
{\sinh \nu (\pi i-\b)}\non \\
&c(\b)=R_0(\b)\frac {\sinh \nu\pi i}{\sinh \nu(\pi i -\b)} \non
\\
& R_0(\b)=\exp
\left\{
i\int\limits _0^{\infty}
\frac {
\sin (\b k)\sinh\frac {\pi k(\nu -1)}{2\nu}
}
{
k\sinh\frac {\pi k}{2\nu}\cosh \frac{\pi k}{2 }
}
\right\}
\non
\end{align}
The coupling constant $\nu $ will be often omitted from $R(\b ,\nu)$.
The relation between R-matrix and XXZ-model is explained later.
From the point of view of mathematics the R-matrix (\ref{R-m}) is
the R-matrix for two-dimensional evaluation representations of the
quantum affine algebra $U_q(\widehat{sl}_2)$. The latter algebra
contains two sub-algebras $U_q(sl_2)$. Let us perform a gauge
transformation with the R-matrix in order to make the invariance with
respect to one of them transparent:
\begin{align}
&\mathcal{R}(\b_1,\b_2,\nu)=
e^{\frac {\nu} 2\b _1\si ^3}\otimes e^{\frac {\nu} 2\b _2\si ^3}\
R(\b_1-\b _2,\nu)
\ e^{-\frac {\nu} 2\b _1\si ^3}\otimes e^{-\frac {\nu} 2\b _2\si ^3}=
\non\\&=
\frac {R_0(\b_1-\b_2)}{2\sinh \nu (\pi i-\b_1+\b_2)}
\(e^{\nu (\b_1-\b _2)}
R_{21}^{-1}(q)-e^{\nu (\b_2-\b _1)}R_{12}(q)\)
\end{align}
where
$$q=e^{2i\pi(\nu+1)}$$
Adding $1$ to $\nu$ is important since we will use fractional powers of $q$.
Here $R(q)$ is usual R-matrix for $U_q(sl_2)$:
$$
R_{12}(q)=
\left(
\begin{array}{cccc}
q^{\frac1 2}&0&0&0\\
0&1&q^{\frac1 2}-q^{-\frac1 2}&0\\
0&0&1&0\\
0&0&0&q^{\frac1 2}
\end{array}
\right)
$$
We want to use this quantum group symmetry. Unfortunately,
the Hamiltonian (\ref{hxxz}) is not invariant with respect to the action of
the quantum group which is represented in the space $\(\mathbb{C}^2\)^{\otimes N}$
by
\begin{align}
&S^3=\sum\limits _{k=1}^N \si ^3_k\non\\
&S^{\pm}=\sum\limits _{k=1}^N q^{-\frac{\si _1^{3}}4}\cdots q^{-\frac{\si_{k-1} ^{3}}4}
\si ^{\pm}_kq^{\frac{\si_{k+1} ^{3}}4}\cdots q^{\frac{\si_{N} ^{3}}4}\non
\end{align}
A solution of this problem of quantum group invariance was
found by Pasquier and Saleur \cite{ps}. They proposed to consider another
integrable model on the finite lattice with Hamiltonian corresponding to
open boundary conditions:
\begin{align}
H_{RXXZ}=\sum\limits _{k=1}^{N-1}(\si ^1_k\si _{k+1}^1+\si ^2_k\si _{k+1}^2+
\Delta\si ^3_k\si _{k+1}^3)
+i\sqrt{1-\Delta}\ (\si _1^3-\si _N^3)
\label{hrxxz}
\end{align}
This Hamiltonian is manifestly invariant under the action of quantum group
on the finite lattice. After the thermodynamic limit one obtains a model
with the same spectrum as original XXZ, but different scattering (this point will be
described later). The infrared limit corresponds to CFT with modified energy-momentum
tensor of central charge
$$c=1-\frac {6\nu ^2}{1-\nu}$$
especially interesting when $\nu$ is rational and additional restriction takes place.
In the present paper we shall consider RXXZ-model.
We shall propose formulae for correlators
for this model showing their similarity with correlators for XXX-model.
The latter can be expressed in terms of values of Riemann
zeta-function at odd natural arguments. We shall obtain an analogue
of this statement for RXXZ-model.
Let us say few words about hypothetic relation of XXZ and RXXZ models
in thermodynamic limit. The argument that this limit should not depend
on the boundary conditions must be dismissed in our situation since
we consider a critical model with long-range correlations. Still we would
expect that the following relation between two models in infinite volume exists.
The quantum group $U_q(sl_2)$ acts on infinite XXZ-model and commute
with the Hamiltonian.
Consider a projector $\mathcal{P}$ on the invariant subspace.
We had XXZ-vacuum $|\text{vac}\rangle _{XXZ}$.
We suppose that the
RXXZ-model is obtained by
projection, in particular:
$$|\text{vac}\rangle _{RXXZ}=\mathcal{P}|\text{vac}\rangle _{XXZ}$$
The correlators in RXXZ-model are
$$
{\ }_{RXXZ}\langle \text{vac}|\mathcal{O}|\text{vac}\rangle _{RXXZ}=
{\ }_{XXZ}\langle \text{vac}|\mathcal{P}\mathcal{O}\mathcal{P}|\text{vac}\rangle _{XXZ}
$$
which can be interpreted in two ways: either as correlator in RXXZ-model
or as correlator of $U_q(sl_2)$-invariant operator
$\mathcal{P}\mathcal{O}\mathcal{P}$ in XXZ-model.
This assumption explains the notation RXXZ standing for Restricted XXZ-model.
So, we assume that in the lattice case a phenomenon close to the
one taking place in massive models occurs \cite{rs}.
Let us explain in some more details the set of operators in XXZ model
for which we are able to calculate the correlators in simple form provided
the above reasoning holds. Under $\mathcal{O}$ we understand
some local operator of XXZ-chain, i.e. a product of several local
spins $\si ^a_k$, $a=1,2,3$. Under the above action of quantum group these
spins transform with respect to 3-dimensional adjoint representation.
The projection $\mathcal{P}\mathcal{O}\mathcal{P}$ extracts
all the invariant operators, i.e. projects over the subspace of singlets
in the tensor product of 3-dimensional representations.
Let us explain more explicitly the relation between the the R-matrix and
XXZ, RXXZ Hamiltonians. Both of them can be constructed form the
transfer-matrix with different boundary conditions constructed via the
monodromy matrix:
$$ R_{01}(\la)R_{02}(\la )\cdots R_{0,N-1}(\la )R_{0,N}(\la )$$
In some cases it is very convenient to consider inhomogeneous model
for which the monodromy matrix contains a fragment:
$$R_{0k}(\la -\la _k)\cdots R_{0,k+n}(\la -\la _{k+n})$$
As we shall see many formulae become far more transparent for
inhomogeneous case.
\section{QKZ on level -4 and correlators.}
The main result of Kyoto group \cite{jmmn,jm} is that the correlators in XXZ-model are
related to solutions of QKZ-equations \cite{SKyo,FR} on level -4. We formulate the
equations first and then explain the relation.
The equations for
the function $g(\b _1,\cdots,\b _{2n})\in \mathbb{C}^{\otimes 2n}$
are
\begin{align}
&R(\b_j -\b _{j+1})g(\b _1,\cdots ,\b _{j+1},\b _j,\cdots,\b _{2n})
=&\non\label{symm}\\
&=
\ \ g(\b _1,\cdots ,\b _{j},\b _{j+1},\cdots,\b _{2n})
%\nonumber\\&\ \non\\
\end{align}
\begin{align}
&g(\b _1,\cdots ,\b _{2n-1},\b _{2n}+2\pi i)
=%\non\\&=
g(\b _{2n},\b _1,\cdots ,\b _{2n-1})
%\non
\label{Rie}
\end{align}
For application to correlators a particular solution is needed which satisfies
additional requirement:
\begin{align}
&g(\b _1,\cdots ,\b _{j},\b _{j+1},\cdots,\b _{2n})|_{\b _{j+1}=\b _j-\pi i}=
s_{j,j+1}\otimes g(\b _1,\cdots ,\b _{j-1},\b _{j+2},\cdots,\b _{2n})\label{norm}
\end{align}
where $s_{j,j+1}$ is the vector $(\uparrow\downarrow)+(\downarrow\uparrow)$
in the tensor product of $j$-th and $(j+1)$-th spaces.
The relation of these equations to correlators is conjectured by
Jimbo and Miwa \cite{jm}. It cannot be proved for critical model
under consideration as it was done for the XXZ-model with $|q|<1$ in \cite{jmmn}.
However, later arguments based on Bethe Anzatz technique were
proposed by Maillet and collaborators \cite{Maillet1,Maillet2} which
can be considered as a proof of Jimbo and Miwa conjecture.
Jimbo and Miwa find the solution needed \cite{jm} in the form:
\begin{align}
&g(\b _1,\cdots, \b _{2n})=\frac 1 {\sum e ^{\b_j}}
\prod\limits _{i