Content-Type: multipart/mixed; boundary="-------------0209301215557" This is a multi-part message in MIME format. ---------------0209301215557 Content-Type: text/plain; name="02-405.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-405.comments" Emptiness formation probability is simpliest of all correlations. We discovered that quantum Knizhnik- Zamolodchikov equations permit to calculate correlations very quikly. We used this in order to support our our conjecture that correlations in XXX spin chain can be expressed in temrs of values of Riemann zeta at odd arguments, ln2 and rational coefficients. ---------------0209301215557 Content-Type: text/plain; name="02-405.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-405.keywords" quantum spin chains, correlation functions, Knizhnik-Zamolodchikov eqaution, Riemann zeta at odd arguments ---------------0209301215557 Content-Type: application/x-tex; name="paper.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="paper.tex" \documentstyle[12pt]{article} \topmargin -.5cm \textheight 23cm \textwidth 180mm \hoffset -20mm \makeatletter \def\eqnarray{\stepcounter{equation}\let\@currentlabel=\theequation \global\@eqnswtrue \global\@eqcnt\z@\tabskip\@centering\let\\=\@eqncr $$\halign to \displaywidth\bgroup\@eqnsel\hskip\@centering $\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne \hfil$\displaystyle{\hbox{}##\hbox{}}$\hfil &\global\@eqcnt\tw@ $\displaystyle\tabskip\z@ {##}$\hfil\tabskip\@centering&\llap{##}\tabskip\z@\cr} \def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth \def\@svsec{}\else \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname.\hskip 1em }\fi \@tempskipa #5\relax \ifdim \@tempskipa>\z@ \begingroup #6\relax \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par} \endgroup \csname #1mark\endcsname{#7}\addcontentsline {toc}{#1}{\ifnum #2>\c@secnumdepth \else \protect\numberline{\csname the#1\endcsname}\fi #7}\else \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname {#7}\addcontentsline {toc}{#1}{\ifnum #2>\c@secnumdepth \else \protect\numberline{\csname the#1\endcsname}\fi #7}}\fi \@xsect{#5}} \def\label#1{\@bsphack\if@filesw {\let\thepage\relax \xdef\@gtempa{\write\@auxout{\string \newlabel{#1}{{\thesection.\@currentlabel}{\thepage}}}}}\@gtempa \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack} \def\@eqnnum{(\thesection.\theequation)} \def\section{\setcounter{equation}{0} \@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\Large\bf}} \newcount\@minsofar \newcount\@min \newcount\@cite@temp \def\appendixname{Appendix} \def\appendix{\par \def\pre@section{\appendixname{}} \setcounter{section}{1} \@addtoreset{equation}{section} \def\thesection{\Alph{section}} \def\theequation{\arabic{equation}}} \def\appendix{\par \def\pre@section{\appendixname{}} \setcounter{section}{1} \@addtoreset{equation}{section} \def\thesection{\Alph{section}} \def\theequation{\arabic{equation}}} \makeatother \def\({\left(} \def\){\right)} \def\[{\left[} \def\]{\right]} \def\non{ \nonumber } \def\z{\zeta} \def\ga{\gamma} \def\debut{\begin{align}} \def\fin{\end{align}} \def\half{\textstyle{\frac 1 2}} \def\ihalf{\textstyle{\frac i 2}} \def\l{\lambda} \def\e{\epsilon} \def\b{\beta} \def\d{\delta} \def\g{\gamma} \def\a{\alpha} \def\s{\sigma} \def\t{\tau} \def\th{\theta} \def\l{\lambda} \def\la{\lambda} \def\e{\epsilon} \def\r{\rho} \def\wid{\widehat} \def\ds{\displaystyle} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\beq{\begin{eqnarray}} \def\eeq{\end{eqnarray}} \def\ov{\overline} \def\om{\omega} \begin{document} \phantom{a} \vspace{0.5cm} \begin{center} {\bf Emptiness Formation Probability and Quantum Knizhnik-Zamolodchikov Equation} \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 one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of formation of a ferromagnetic string $P(n)$ in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of the EFP in the inhomogeneous case. It is based on the quantum Knizhnik-Zamolodchikov equation [qKZ]. We calculate EFP for $n\le 6$ for inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations and number theory. We also make a conjecture about a structure of EFP for arbitrary $n$. \end{abstract} \newpage \section{Introduction} The Hamiltonian of the XXX Heisenberg spin chain can be written like this \be H = \sum_{i=-N}^N \> (\s^x_i\s^x_{i+1}\; + \;\s^y_i\s^y_{i+1}\; +\; \s^z_i\s^z_{i+1}\;-1\;) \label{H} \ee Here $2 N +1 $ is the length of the lattice and $\s^x_i,\s^y_i,\s^z_i$ are Pauli matrices. We consider thermodynamic limit [ $N$ goes to infinity] . The sign in front of the Hamiltonian indicates that we are dealing with the antiferromagnetic case. We also imply periodic boundary conditions. The model was solved by Bethe in 1931, see \cite{B}. The ground state was constructed by by Hulth\'{e}n, see \cite{H}. We shall denote the ground state in the thermodynamic limit by ${| {\rm GS} \rangle}$ . The emptiness formation probability (EFP) for the XXX model is defined as follows: \begin{equation} P(n) = \langle {\rm GS} | \prod^n_{j=1} P_j | {\rm GS} \rangle, \end{equation} here ${P_j = S^z_j+\half}$ is the projector on the state with the spin up in the ${j}$-th lattice site. The integer $n$ has a meaning of a length of a continuous ferromagnetic string. $P(n)$ is a probability that this string can appear in the antiferromagnetic ground-state. The importance of EFP was emphasized in \cite{KBI}. $P(n)$ can be represented as $n$-multiple integral. The integral representation follows from the work of RIMS group. RIMS approach is based on vertex operators and bosonic representation of infinite-dimensional quantum algebras, see \cite{JMN,JM}. The explicit formula for $P(n)$ in XXX limit was obtained in \cite{KIEU}: %For the homogeneous case the integral representation for $P(n)$ %follows as a limit from the formulae for the correlation functions %which were obtained by the RIMS group with the help of %the vertex operator approach and the bosonic %representation of infinite-dimensional quantum algebras %\cite{JMN,JM}. %The explicit formula was obtained by Essler, Izergin, Korepin and %Uglov \cite{KIEU}: \be P(n)= \prod_{j=1}^n\int_{C}{d\lambda_j\;\over 2\pi i }\; U_n(\l_1,\ldots,\l_n)\;T_n(\l_1,\ldots,\l_n) \label{int} \ee where \be U_n(\l_1,\ldots,\l_n)\;=\;\pi^{{n(n+1)\over 2}}\> {\prod_{1\leq k < j\leq n}\sinh{\pi(\lambda_j-\lambda_k)} \over \prod_{j=1}^n\sinh^n{\pi\lambda_j} } \label{U_n} \ee and \be T_n(\l_1,\ldots,\l_n)\;=\; {\prod_{j=1}^n\l_j^{j-1}(\lambda_j+i)^{n-j}\over \prod_{1\leq k < j\leq n}(\lambda_j-\lambda_k-i) } \label{T_n} \ee The contour $C$ goes parallel to the real axis with the imaginary part confined between $0$ and $-i$ for each integral. In papers \cite{BK1,BK2,BKNS} we % have worked out a technique by means of which we could evaluated the integrals %(\ref{int}) for $n\le 5$. We discovered that these $P(n)$ can be expressed in terms of the values of Riemann zeta function at odd arguments, $\log 2$ and rational coefficients. We conjectured that this is a general property for all $P(n)$. In this paper we proved the property for $P(6)$. We think that all correlation functions \be \langle {\rm GS} | \s^z_{i_1}\;\s^z_{i_2}\;\ldots \s^z_{i_m}| {\rm GS} \rangle \label{cor} \ee also have this property. %which is an average of any number of operators $\s^z$ attached to %some lattice sites. Asymptotic behavior of $P(n)$ for large $n$ was studied in the papers \cite{BK1},\cite{BKNS}, \cite{AK} and \cite{LNKS}. The technique of calculation of these integrals, described in the paper \cite{BK2} worked for $n=2,3,4$. % For larger $n$ it is more difficult. %Unfortunately, an application of the technique described in detail %in \cite{BK2} for %the evaluation of $P(n)$ when $n=2,3,4$ gets rather complicated %with the growth of $n$. In the paper \cite{BKNS} we calculated $P(5)$ by means of this technique. However, these computations are so complicated that it is problematic to generalize them to the case $n=6$. % not speaking about the higher $n$ case. So we start looking for indirect methods of evaluation of the integrals (\ref{int}). %relatively quickly evaluate the integrals like (\ref{int}). It appeared to be useful to consider inhomogeneous case. %instead of the homogeneous one In this case there are more free parameters. We call them inhomogeneity parameters and denote by $z_1,\ldots,z_n$. The EFP in the inhomogeneous case we shall denote by $P_n(z_1,\ldots,z_n)$. Let us remind to the reader that inhomogeneous models were used for evaluation of correlation functions from the very beginning. For the massive regime of the XXZ model the vertex operator approach was developed in \cite{JMN,JM}. It allowed to express the correlation functions in terms of the trace functions. Special combinations of these trace functions satisfy the quantum Knizhnik-Zamolodchikov equation (qKZ), see \cite{KZ,FR}. Later in paper \cite{JM1} Miwa and Jimbo suggested that the correlation functions in the gap-less regime directly satisfy the qKZ equation. Since the XXX model belongs to the gap-less regime we shall use qKZ for evaluation of EFP in the inhomogeneous case. We suggest a general ansatz for $P_n(z_1,\ldots,z_n)$, see (3.20). This constitute a new method for computation of the EFP. On the other hand, it is easy to generalize the technique explained in \cite{BK2} to the inhomogeneous case and calculate the EFP directly [for short ferromagnetic strings]. When we can compare results, they coincide. The paper is organized as follows. In the Section 2 we discuss the relation of EFP to the qKZ and derive three important properties of the EFP in the inhomogeneous case. % which we denote $P_n(z_1,\ldots,z_n)$ with inhomogeneity parameters $z_1,\ldots,z_n$ or shortly % $P_n$. In Section 3 we apply a generalization of the technique described in \cite{BK2} to the inhomogeneous case and compute $P_n$ directly for $n\le 4$. Then we check that these $P_n$ satisfy all the properties, which follow from qKZ. This helps us to formulate a general {\bf ansatz} for $P_n$ in the inhomogeneous case. Further, we suggest a new way of computing $P_n$. One can use the ansatz and general properties of $P_n$ [which follow from the qKZ]. In this way we get the explicit expressions for $P_5$ and $P_6$ in the inhomogeneous case. In Section 4 we discuss the homogeneous limit of $P(n)$ for $n\le 6$. In particular, when $n\le 5$ we reproduce our previous results, obtained in \cite{BK1,BK2,BKNS}. We also get the analytic expression for $P(6)$ in the homogeneous limit. Having this answer we can compare it with the numerical value for $P(6)$ obtained by the DMRG method in \cite{BKNS}. We also discuss the structure of EFP in the homogeneous case and offer some plausible conjectures. In the last Section 6 we discuss the results and outline some possible ways of a further progress. \section{The EFP in the inhomogeneous case and the qKZ equation.} We believe that consideration of the inhomogeneous case instead of the homogeneous one can give us a new information about the EFP and other correlation functions. A method of calculation of correlation functions, which we use was found in the papers \cite{JMN, JM}. It is based on theory of infinite-dimensional quantized algebras and vertex operators. We shall need elements of this method. Let us introduce some notations. We use the R-matrix: \be R(\l)= \frac { R_0(\l)} {\l+\pi i} \left( \begin{array}{cccc} \l+\pi i&0&0&0\\ 0&-\l&\pi i&0\\ 0&\pi i&-\l&0\\ 0&0&0&\l+\pi i \end{array} \right) \label{R-m} \ee where $$ R_0(\l)=-\frac {\Gamma\(\frac \l {2\pi i}\)\Gamma\(\frac 1 2-\frac \l {2\pi i}\)} {\Gamma\(-\frac \l {2\pi i}\)\Gamma\(\frac 1 2+\frac \l {2\pi i}\)}, $$ notice that $$R_0(\l)R_0(-\l)=1$$ This R-matrix appears in the rational limit from XXZ R-matrix, it is related to usual XXX R-matrix by $$R(\l)=(\sigma ^3\otimes I )R_{XXX}(\l)(I\otimes \sigma ^3)$$ (similar transformation is needed when obtaining form factors of $SU(2)$-invariant Thirring model from SG ones \cite{book}). This R-matrix (\ref{R-m}) satisfies the equation: \be R(-\pi i)= \left( \begin{array}{cccc} 0&0&0&0\\ 0&1&1&0\\ 0&1&1&0\\ 0&0&0&0 \end{array} \right) \label{proj} \ee Following \cite{JM} we introduce functions $g_n$ which satisfy the qKZ equations on level $-4$ \cite{FR}. We write the qKZ equations in their original form \cite{book,sm} which takes into account explicitly symmetry: \beq &g_n(\l _1,\cdots ,\l _{j+1},\l _j,\cdots,\l _{2n})_ {\e _1,\cdots ,\e _{j+1}',\e _j',\cdots,\e _{2n}}=&\label{symm}\\ &=R(\l _j -\l _{j+1})^{\e _{j},\e _{j+1}}_{\e _{j+1}',\e _j'} \ \ g_n(\l _1,\cdots ,\l _{j},\l _{j+1},\cdots,\l _{2n}) _{\e _1,\cdots ,\e _{j},\e _{j+1},\cdots,\e _{2n}} \nonumber \eeq \be g_n(\l _1,\cdots ,\l _{2n-1},\l _{2n}+2\pi i)_ {\e _1,\cdots ,\e _{2n-1},\e _{2n}} = g_n(\l _{2n},\l _1,\cdots ,\l _{2n-1}) _{\e _{2n},\e _1,\cdots ,\e _{2n-1}} \label{Rie} \ee Solutions to these equations are meromorphic functions with possible singularities at the points $$ \Im (\l _j-\l_k)=\pi l, \quad l\in {\mbox{\bf Z}}\backslash 0$$ For application to the correlation functions we are interested in a particular solution $g_n$ described in details in \cite{JMN,JM}. Detailed study of this solution will be performed in a future publication, in the present paper we need only limited information about it. First, the $g_n$ is regular at $ \Im (\l _j-\l_k)=\pm\pi $. Moreover, much can be said about its values at these points \cite{JM} : \beq &g_n(\l _1,\cdots ,\l _{j-1},\l _{j},\l _j-\pi i,\l _{j+2},\cdots,\l _{2n})_ {\e _1,\cdots ,\e _{j-1},\e _{j},\e _{j+1},\e _{j+2},\cdots,\e _{2n}}=& \nonumber\\ &=\delta_{\e _{j},-\e _{j+1}} \ g_{n-1}(\l _1,\cdots ,\l _{j-1},\l _{j+2},\cdots,\l _{2n}) _{\e _1,\cdots,\e _{j-1},\e _{j+2}, \cdots,\e _{2n}} &\label{val1} \eeq Together with the symmetry (\ref{symm}), and (\ref{proj}) this equation implies \beq &\sum\limits_{\e _j=-\e _{j+1}}g_n(\l _1,\cdots ,\l _{j-1}, \l _{j},\l _j+\pi i,\l _{j+2},\cdots,\l _{2n})_ {\e _1,\cdots ,\e _{j-1},\e _{j},\e _{j+1},\e _{j+2},\cdots,\e _{2n}} %+\non\\&+ %g_n(\l _1,\cdots ,\l _{j},\l _j+\pi i,\cdots,\l _{2n})_ %{\e _1,\cdots ,-,+,\cdots,\e _{2n}} =&\nonumber\\ &= g_{n-1}(\l _1,\cdots ,\l _{j-1},\l _{j+2},\cdots,\l _{2n}) _{\e _1,\cdots,\e _{j-1},\e _{j+2}, \cdots,\e _{2n}} &\label{val2} \eeq The emptiness formation probability $P_n$ is related to $g_n$ as follows \be P_n(z_1,\cdots ,z_n)= g_n\(\pi z_1,\cdots, \pi z_n,\pi (z_n+i),\cdots,\pi (z_1+i)\) _{-,\cdots ,-,+,\cdots ,+} \label{Png} \ee Now we want to establish some general properties of $P_n$ following from (\ref{symm}), (\ref{Rie}), (\ref{val1}), (\ref{val2}). \noindent 1. Symmetry. {\it The function $P_n(z_1,\cdots ,z_n)$ is symmetric.} \noindent{\it Proof.} Obviously, it is enough to show that $$P_n(\cdots,z_j,z_{j+1},\cdots)=P_n(\cdots,z_{j+1},z_j,\cdots)$$ This identity follows from (\ref{symm}) and from the fact that the R-matrix acts diagonally on the indices $-,-$ and $+,+$: \beq &P_n(\cdots,z_{j},z_{j+1},\cdots)=&\nonumber\\ &= g_n\(\cdots,\pi z_{j},\pi z_{j+1},\cdots,\pi (z_{j+1}+i),\pi(z_j+i)\cdots\) _{\cdots, -,-,\cdots ,+,+,\cdots}=&\nonumber\\ &= R_0(\pi (z_{j+1}-z_j))R_0(\pi (z_{j}-z_{j+1}))&\nonumber\\ &\times g_n\(\cdots,\pi z_{j+1},\pi z_{j},\cdots,\pi (z_{j}+i),\pi(z_{j+1}+i)\cdots\) _{\cdots, -,-,\cdots ,+,+,\cdots}=\non\\&=P_n(\cdots,z_{j+1},z_{j},\cdots) &\nonumber \eeq \hfill{\bf QED} \noindent 2. Vanishing. {\it The function $P_n(z_1,\cdots ,z_n)$ vanishes when $z_k=z_j+i$.} \noindent{\it Proof.} Due to the previous property it is sufficient to consider the case $k=1$, $j=2$. Let us put, first, $z_1=z+i$, $z_{2}=z'$, then we shall take the limit $z\to z'$. \beq &P_n(z+i,z',\cdots ,z_n)=&\nonumber\\ &= g_n\(\pi (z+i),\pi z',\cdots, \pi z_n,\pi (z_n+i),\cdots,\pi(z'+i),\pi (z+2i)\) _{-,-,\cdots, -,+,\cdots ,+,+}=&\nonumber\\ &= g_n\(\pi z,\pi (z+i),\pi z',\cdots, \pi z_n,\pi (z_n+i),\cdots,\pi(z'+i),\) _{+,-,-,\cdots, -,+,\cdots ,+}&\nonumber \eeq where we used (\ref{Rie}). Consider the limit $z\to z'$. As it has been explained singularities do not occur for $\Im (\l _j -\l _k)=\pm \pi$. Moreover, the final result contains the fragment $$g_n(\cdots, \pi (z+i),\pi z\cdots)_{\cdots ,-,-,\cdots }$$ which implies that the result vanishes due to (\ref{val1}). Because of absence of singularities this zero does not interfere with any pole, so \be P_n(z+i,z,\cdots ,z_n)=0 \label{vanish} \ee \hfill{\bf QED} \noindent 3. Normalization. {\it The following asymptotic holds for $z_1\to\infty$ along the real axis:} \be P_n(z_1,z_2,\cdots,z_n)\to \half P_{n-1}(z_2,\cdots,z_{n})\label{norm} \ee \noindent{\it Proof.} One more property of the solution $g_n$ will be important for us. Using the integral formula from \cite{JM} one can show that $g_n(\l_1,\cdots ,\l _{2n-2},\l,\l+\pi i)$ behaves as $O(1)$ when $\l\to\infty +i\kappa$ where $\kappa$ is a finite number. The leading term of asymptotic does not depend on $\kappa$. We shall use notation: \be g_n(\l_1,\cdots ,\l _{2n-2},\l,\l+\pi i)_ {\e_1,\cdots ,\e_{2n-2},\e_{2n-1},\e_{2n}}\to \widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ \e_{2n-1},\e_{2n}} \label{hat} \ee The function $\widehat{g}_n$ possesses important property of symmetry with respect to last two indices because \beq &g_n(\l_1,\cdots ,\l _{2n-2},\l,\l+\pi i)_{\e_1,\cdots ,\e_{2n-2},\e_{2n-1},\e_{2n}}=& \nonumber\\ &= g_n(\l-\pi i,\l_1,\cdots ,\l _{2n-2},\l)_{\e_{2n}\e_1,\cdots ,\e_{2n-2},\e_{2n-1}}& \nonumber\\ &= R(\l_1-\l+\pi i) _{\e_1,\e_{2n}}^{\e_1',\sigma _{2n-3}} \cdots R(\l_{2n-2}-\l+\pi i)_{\e_{2n-2},\sigma_1}^{e_{2n-2}',\e _{2n}'}&\nonumber\\ &\times g_n(\l_1,\cdots ,\l _{2n-2},\l-\pi i,\l)_{\e_1',\cdots , \e_{2n-2}',\e_{2n}',\e_{2n-1}}\rightarrow &\nonumber\\ & \rightarrow \mbox{sign}(\e_{2n-1},\e_{2n})\, \widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ \e_{2n},\e_{2n-1}}& \label{ghat} \eeq with the sign function \be \mbox{sign}(\e_1,\e_2)\;=\; \cases{ {\ds -1 } & if ${\ds \e_1=\e_2}$ \cr {\ds \>1 } & if ${\ds \e_1=-\e_2}$ \cr } \ee where we used the asymptotic of the R-matrix $$R(\l)\ \longrightarrow {\hskip -.8cm}_{_{\l\to\infty}}\ (-i)\cdot\mbox{diag}(1,-1,-1,1)$$ From the equation (\ref{ghat}) we conclude that $$ \widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ \e_{2n-1},\e_{2n}}=0 \quad \mbox{if} \quad \e_{2n-1}=\e_{2n} $$ \be \widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ +,-}= \widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ -,+} \label{ghat+-} \ee The relation (\ref{ghat+-}) allows to calculate $\widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ +,-}$: \beq &g_{n-1}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}}=&\nonumber\\ &= \widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ +,-}+%\non\\&+ \widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ -,+}=& \nonumber\\ &= 2\ \widehat{g}_{n}(\l_1,\cdots ,\l _{2n-2})_{\e_1,\cdots ,\e_{2n-2}\ ;\ +,-}& \nonumber \eeq Now the normalization (\ref{norm}) follows from \beq &P_n(z_1,z_2,\cdots ,z_n)=&\nonumber\\ &= g_n\(\pi z_1,\pi z_2,\cdots, \pi z_n,\pi (z_n+i),\cdots,\pi (z_2+i),\pi (z_1+i)\) _{-,-,\cdots, -,+,\cdots ,+,+}=&\nonumber\\ &= g_n\(\pi z_2,\cdots, \pi z_n,\pi (z_n+i),\cdots,\pi (z_2+i),\pi (z_1+i),\pi (z_1+2i)\) _{-,\cdots, -,+,\cdots ,+,+,-}&\nonumber\\ & \longrightarrow\hskip -.8cm _{_{z_1\to\infty}} \ \widehat{g}_n\(\pi z_2,\cdots, \pi z_n,\pi (z_n+i),\cdots,\pi (z_2+i)\) _{-,\cdots, -,+,\cdots ,+\ ;\ +,-}=&\nonumber\\ &= \half \ P_{n-1}(z_2,\cdots ,z_n)& \eeq where the formula (\ref{hat}) was used. \hfill{\bf QED} The above three properties will be very useful for further consideration. \section{Explicit expressions for EFP.} In paper \cite{JM1} Jimbo and Miwa suggested an integral representation as a solution to the qKZ (\ref{symm}) - (\ref{Rie}) which also satisfies the property (\ref{val1}). Using the relation (\ref{Png}) one gets the integral representation which is the direct generalization of the formula (\ref{int}) to the inhomogeneous case $$ P_n(z_1,\ldots,z_n) = \pi^{{n(n+1)\over 2}}\prod_{k0}{(-1)^{n-1}\over n^s}\;=\; - \mbox{Li}_s(-1) \label{za} \ee where $\mbox{Li}_s(x)$ is the polylogarithm. The alternating zeta series is related to the Riemann zeta function as follows \be \zeta(s)\;=\;{1\over 1-2^{1-s}}\zeta_a(s) \label{za1} \ee Unlike Riemann zeta function the alternating zeta series is regular at $s=1$, it is $\zeta_a(1)=\log 2$. Let us mention that the answers for $P(1),\ldots, P(4)$ can be obtained from the formulae (\ref{P1}-\ref{P4}) by taking the homogeneous limit i.e. $z_j\rightarrow 0$ and using the expansion of the function $G$ (\ref{Gexp}). The same can be also done for $n=5$ and $n=6$ by means of the formula (\ref{Pninhom}), formulae (\ref{An0}), (\ref{Anl}), (\ref{Q}) and the answers for the polynomials (\ref{Q52res}-\ref{Q51res}) and (\ref{Q63res}-\ref{Q61res}) for the cases $n=5$ and $n=6$ respectively. For $n\le 5$ we reproduce the known results, see \cite{BKNS} formula (1.16) in there. In the case $n=6$ we discover a new result \beq &\ds P(6)\; =\; {1\over 7} \biggl\{ 1 - 35\,\zeta_a(1) + 322\,\zeta_a(3) - {9244\over 5}\,\zeta_a(5) + {22694\over 5}\,\zeta_a(7) - 2982\,\zeta_a(9) &\nonumber\\ &\ds - {3920\over 3}\,\zeta_a(1)\cdot\zeta_a(3) + {369908\over 15}\,\zeta_a(1)\cdot\zeta_a(5) - {28784\over 5}\,\zeta_a(3)^2 - {263816\over 3}\,\zeta_a(1)\cdot\zeta_a(7) &\nonumber\\ &\ds + {3458\over 15}\,\zeta_a(3)\cdot\zeta_a(5) +{323344\over 5}\,\zeta_a(1)\cdot\zeta_a(9) + {933702\over 5}\,\zeta_a(3)\cdot\zeta_a(7) - {751592\over 9}\,\zeta_a(5)^2 &\nonumber\\ &\ds - {2627842\over 15}\,\zeta_a(3)\cdot\zeta_a(9) +{235963\over 9}\,\zeta_a(5)\cdot\zeta_a(7) +{368564\over 3}\,\zeta_a(5)\cdot\zeta_a(9) - {644987\over 9}\,\zeta_a(7)^2 &\nonumber\\ &\ds +{538496\over 45}\,\zeta_a(1)\cdot\zeta_a(3)\cdot\zeta_a(5) -{269248\over 135}\,\zeta_a(3)^3 -{1143268\over 9}\,\zeta_a(1)\cdot\zeta_a(3)\cdot\zeta_a(7) + {653296\over 9}\,\zeta_a(1)\cdot\zeta_a(5)^2 &\nonumber\\ &\ds -{163324\over 45}\,\zeta_a(3)^2\cdot\zeta_a(5) +{1737148\over 15}\,\zeta_a(1)\cdot\zeta_a(3)\cdot\zeta_a(9) -{1737148\over 45}\,\zeta_a(3)^2\cdot\zeta_a(7) +{124082\over 9}\,\zeta_a(3)\cdot\zeta_a(5)^2 &\nonumber\\ &\ds - {528164\over 3}\,\zeta_a(1)\cdot\zeta_a(5)\cdot\zeta_a(9) + {924287\over 9}\,\zeta_a(1)\cdot\zeta_a(7)^2 + {264082\over 5}\,\zeta_a(3)^2\cdot\zeta_a(9) &\nonumber\\ &\ds -{264082\over 9}\,\zeta_a(3)\cdot\zeta_a(5)\cdot\zeta_a(7) +{188630\over 27}\,\zeta_a(5)^3 \biggr\} &\label{P6} \eeq This support our hypothesis that all $P(n)$ can be expression in terms of values of Riemann zeta functions at odd arguments, $\log 2$ and rational coefficients. From this expression we can get numerical value \be P(6) = 7.0685928\cdot 10^{-9} \label{number} \ee In \cite{BKNS} a numerical method was used for evaluation of $P(6)$. It is called Density Matrix Renormalization Group (DMRG). The results can be found in Table 1 in \cite{BKNS}. In particular $P(6)=7.05\cdot 10^{-9}$ with an uncertainty in the second digit after the decimal point. It is in a good agreement with our analytic result (\ref{P6}), (\ref{number}). Looking at the formulae (1.16) of \cite{BKNS} and the above expression (\ref{P6}) we can make a general conjecture for the dependence of $P(n)$ on the alternating zeta series \be P(n)\;=\;{1\over (n+1)}\sum_{{\vec r}\in U} B^{(n)}_{r_0,r_1,\ldots,r_{n-2}} \prod_{j=0}^{n-2} {[\zeta_a(2\,j+1)]}^{r_j} \label{Pn} \ee where all coefficients $B^{(n)}_{r_0,r_1,\ldots,r_{n-2}}$ are rational and the sum is over non-negative integers $r_0,\ldots,r_{n-2}$ which belong to the region $U$ determined by the following two conditions \beq %&\ds 0\le r_j\le \mbox{min}(j+1,n-j-1), \quad j=0, 1,\ldots, n-2 & \nonumber\\ &\ds \sum_{j=0}^{n-2} r_j\;\le [n/2] & \nonumber\\ &\ds \sum_{j=0}^{n-2} r_j\,(2j+1)\;\le \frac{n(n-1)}{2}& \label{U} \eeq Let us show how the non-zero coefficients $B^{(n)}_{r_0,r_1,\ldots,r_{n-2}}$ look like for the cases when we know the manifest analytic answer, namely, when $n=1,2,\ldots 6$ \beq & B^{(1)}\;=\;1 & \nonumber\\ &\quad &\nonumber\\ & B^{(2)}_{0}\;=\;1\quad B^{(2)}_{1}\;=\;-1 & \nonumber\\ &\quad &\nonumber\\ & B^{(3)}_{0,0}\;=\;1\quad B^{(3)}_{1,0}\;=\;-4 \quad B^{(3)}_{0,1}\;=\;2 & \nonumber\\ &\quad &\nonumber\\ &\ds B^{(4)}_{0,0,0}\;=\;1 \quad B^{(4)}_{1,0,0}\;=\;-10 \quad B^{(4)}_{0,1,0}\;=\;\frac{173}{9} \quad B^{(4)}_{0,0,1}\;=\;-\frac{110}{9} & \nonumber\\ &\ds B^{(4)}_{1,1,0}\;=\;-\frac{110}{9} \quad B^{(4)}_{1,0,1}\;=\;\frac{170}{9} \quad B^{(4)}_{0,2,0}\;=\;-\frac{17}{3} & \nonumber\\ &\quad &\nonumber\\ &\ds B^{(5)}_{0,0,0,0}\;=\;1 \quad B^{(5)}_{1,0,0,0}\;=\;-20 \quad B^{(5)}_{0,1,0,0}\;=\;\frac{281}{3} \quad B^{(5)}_{0,0,1,0}\;=\;-\frac{1355}{6} \quad B^{(5)}_{0,0,0,1}\;=\;\frac{889}{6} & \nonumber\\ &\ds B^{(5)}_{1,1,0,0}\;=\;-180 \quad B^{(5)}_{1,0,1,0}\;=\;\frac{3920}{3} \quad B^{(5)}_{1,0,0,1}\;=\;-\frac{3290}{3} \quad B^{(5)}_{0,1,1,0}\;=\;-\frac{170}{3} \quad B^{(5)}_{0,1,0,1}\;=\;679 & \nonumber\\ &\ds B^{(5)}_{0,2,0,0}\;=\;-326 \quad B^{(5)}_{0,0,2,0}\;=\;-\frac{970}{3} &\nonumber\\ &\quad &\nonumber\\ &\ds B^{(6)}_{0, 0, 0, 0, 0} = 1 &\nonumber\\ &\ds B^{(6)}_{1, 0, 0, 0, 0} = -35\quad B^{(6)}_{0, 1, 0, 0, 0} = 322\quad B^{(6)}_{0, 0, 1, 0, 0} = -{9244\over 5}\quad B^{(6)}_{0, 0, 0, 1, 0} = {22694\over 5}\quad B^{(6)}_{0, 0, 0, 0, 1} = -2982\quad &\nonumber\\ &\ds B^{(6)}_{1, 1, 0, 0, 0} = -{3920\over 3}\quad B^{(6)}_{1, 0, 1, 0, 0} = {369908\over 15}\quad B^{(6)}_{1, 0, 0, 1, 0} = -{263816\over 3}\quad B^{(6)}_{1, 0, 0, 0, 1} = {323344\over 5}\quad &\nonumber\\ &\ds B^{(6)}_{0, 1, 1, 0, 0} = {3458\over 15}\quad B^{(6)}_{0, 1, 0, 1, 0} = {933702\over 5}\quad B^{(6)}_{0, 1, 0, 0, 1} = -{2627842\over 15}\quad B^{(6)}_{0, 0, 1, 1, 0} = {235963\over 9}\quad &\nonumber\\ &\ds B^{(6)}_{0, 0, 1, 0, 1} = {368564\over 3}\quad B^{(6)}_{0, 2, 0, 0, 0} = -{28784\over 5}\quad B^{(6)}_{0, 0, 2, 0, 0} = -{751592\over 9}\quad B^{(6)}_{0, 0, 0, 2, 0} = -{644987\over 9}\quad &\nonumber\\ &\ds B^{(6)}_{1, 1, 1, 0, 0} = {538496\over 45} \quad B^{(6)}_{1, 1, 0, 1, 0} = -{1143268\over 9}\quad B^{(6)}_{1, 1, 0, 0, 1} = {1737148\over 15}\quad B^{(6)}_{1, 0, 1, 0, 1} = -{528164\over 3} \quad &\nonumber\\ &\ds B^{(6)}_{0, 1, 1, 1, 0} = -{264082\over 9}\quad B^{(6)}_{1, 0, 2, 0, 0} = {653296\over 9}\quad B^{(6)}_{1, 0, 0, 2, 0} = {924287\over 9}\quad B^{(6)}_{0, 1, 2, 0, 0} = {124082\over 9}\quad &\nonumber\\ &\ds B^{(6)}_{0, 2, 1, 0, 0} =-{163324\over 45} \quad B^{(6)}_{0, 2, 0, 1, 0} = -{1737148\over 45}\quad B^{(6)}_{0, 2, 0, 0, 1} = {264082\over 5}\quad &\nonumber\\ &\ds B^{(6)}_{0, 3, 0, 0, 0} = -{269248\over 135}\quad B^{(6)}_{0, 0, 3, 0, 0} = {188630\over 27}\quad &\label{B} \eeq For us it was a bit surprising that two of coefficients $B^{(6)}$ appeared to be zero, namely, \be B^{(6)}_{1, 0, 1, 1, 0} = 0 \quad B^{(6)}_{1, 2, 0, 0, 0} = 0 \label{B6zero} \ee It means that the structures $\zeta_a(1)\cdot\zeta_a(5)\cdot\zeta_a(7)$ and $\zeta_a(1)\cdot\zeta_a(3)^2$ do not appear in the final answer for $P(6)$. Meanwhile the term $\zeta_a(3)^3$ survived with the non-zero coefficient $B^{(6)}_{0, 3, 0, 0, 0}$ which we expected to be zero. The most evident conjecture that we can make looking at the formulae (\ref{B}) is as follows \be B^{(n)}_{0,0,\ldots,0}\;=\;1, \label{B0} \ee \be B^{(n)}_{1,0,\ldots,0}\;=\;-\,\left(\begin{array}{c} n+1\\3\end{array}\right) \label{B1} \ee where $\left(\begin{array}{c} n\\m\end{array}\right)$ is the binomial coefficient. The next our conjecture is less trivial. As appeared the coefficients $B$ might satisfy some equations. One of them has the following form \be \sum_{r_0+r_1+\ldots +r_{n-2}\,=\,p} B^{(n)}_{r_0,r_1,\ldots,r_{n-2}}\;=\;(-1)^p \left(\begin{array}{c} n-p\\p\end{array}\right) \label{sumB} \ee where $p$ is some fixed positive integer. The expression (\ref{sumB}) can be easily verified for the first coefficients given by the formulae (\ref{B}). For example, when $p=1$ we get the following equation \be B^{(n)}_{1,0,\ldots,0}\,+\,B^{(n)}_{0,1,\ldots,0}\,+\ldots\,+\, B^{(n)}_{0,0,\ldots,1}\;=\;-\,n\,+\,1 \label{sumB1} \ee where we have already made the conjecture (\ref{B1}) for the first term. We believe that there should be more equations like (\ref{sumB}) which probably provide the rigorous expression for the coefficients $B$. We have pointed out in our previous work \cite{BKNS} that since $\zeta_a(1), \zeta_a(3), \zeta_a(5), \ldots $ are very likely different irrational (or even transcendental) numbers \footnote{It was proven by Ap{\'e}ry \cite{Ap} that $\zeta(3)$ is irrational. Then Rivoal \cite{Riv} proved that one of the nine numbers $\zeta(5),\ldots,\zeta(21)$ is irrational. One of the most recent theorem proved by Zudilin \cite{Zud} says that one of the four values $\zeta(5),\ldots,\zeta(11)$ is irrational. (See also the paper by D. Zagier \cite{za} and the paper by Yu. Nesterenko \cite{Nest}). } $P(n)$ seem to be different irrational (or transcendental) numbers as well. This means that $P(n)$ does not satisfy polynomial recursion relation with respect to the distance. In order to clarify the structure of the formula (\ref{Pn}) let us formally replace all $\zeta_a(2j+1)$ by one complex variable $x$. This will define a new function $P(n,x)$. Then we can calculate $P(n,x)$ using our conjectures (\ref{Pn}), (\ref{sumB}) and some properties of the binomial coefficients. Namely, \be P(n,x)\;=\; \sum_{p=0}^{[n/2]}(-x)^p\left(\begin{array}{c} n-p\\p\end{array}\right)\;=\; {A_+^{n+1}-A_-^{n+1}\over (n+1)\sqrt{1-4 x}} \label{pnx} \ee where \be A_{\pm}\;=\;{1\pm \sqrt{1- 4 x}\over 2} \label{Apm} \ee In particular when $x\rightarrow 1$ one gets \be P(n,1)\;=\;\cases{ \;\; 1, \quad n = 6 k\quad \mbox{or} \quad n = 6 k+1 \cr \;\; 0, \quad n = 3 k+2 \cr -1, \quad n = 6 k+3\quad \mbox{or} \quad 6 k+4 \cr } \label{Pn1} \ee Note that the alternating zeta values approaches $1$ as the argument gos to infinity \be \lim_{s\rightarrow\infty}\zeta_a(s)\;=\;1 \label{limit} \ee Another nice form of the formula (\ref{pnx}) can be obtained by substitution \be x = {1\over {4\cosh^2 {\a}}} \label{xal} \ee Then \be P(n,{1\over 4\cosh^2{\a}})={\sinh{[(n+1)\a]}\over 2^n(n+1)\cosh^n{\a} \sinh{\a}} \label{Pnal} \ee We see that when $\a$ tends to zero \be \lim_{\a\rightarrow 0}P(n,{1\over 4\cosh^2{\a}})={1\over 2^n} \label{Pnal0} \ee Let us remark that this result appeared to coincide with the limiting formula (\ref{inftylimit}). We do not know if this is accidental or there is a reason for this. Let us briefly discuss the generating function for the values $P(n)$ \be \Psi(y)\;=\;\sum_{n=0}^{\infty}y^n \, P(n) \label{gen1} \ee where $P(0)=1$ by definition. Taking into account the conjectures (\ref{B0}) and (\ref{B1}) we can easily get the two first terms for the generating function $\Psi(y)$ \be \Psi(y)\;=\;-{\ln{(1-y)}\over y}\;+\;{y^2\over 3(1-y)^3}\ln{2}\;+\;\ldots \label{Psi} \ee As we discussed in \cite{BK1} and \cite{BKNS} we expect that for $n\gg 1$ \be P(n)\sim e^{-\kappa n^2} \label{assPn} \ee If we substitute it formally into eq. (\ref{gen1}) with $y=e^{u}$ then we can expect that \be \Psi(e^u)\sim \tilde\Psi(u)\;=\;\sum_{n=0}^{\infty} e^{-\kappa n^2 + u\,n} \label{psiu} \ee so that the function $\tilde\Psi(u)$ satisfies the functional equation \be \tilde\Psi(u)\,+\,\tilde\Psi(-u)\,-\,1\;=\;\theta_3({iu\over 2},e^{-\kappa}) \label{funrel} \ee where $\theta_3$ is the third Jacobi theta function with the nome $e^{-\kappa}$. So we can expect that the generating function (\ref{gen1}) may be related with the elliptic functions and have some automorphic properties. Another possibility is to put into the r.h.s. of eq. (\ref{gen1}) the values $P(n,x)$ given by (\ref{pnx}) instead of $P(n)$ then \be \Psi(x,y)\;=\;\sum_{n=0}^{\infty}y^n \, P(n,x)\; =\; {1\over y\sqrt{1-4x}}\ln{2-y+y\sqrt{1-4x}\over 2-y-y\sqrt{1-4x}} \label{gen1x} \ee Another generating function looks much simpler \be \Psi'(x,y)\;=\;\sum_{n=0}^{\infty}(n+1)y^n \, P(n,x)\; =\; {1\over 1\,-\,y\,+\,x y^2} \label{gen1x'} \ee \section{Discussion} The main point of this communication was to consider the emptiness formation probability in the inhomogeneous case. The basic advantage of the inhomogeneous case is that we have more parameters at our disposal and relation to the qKZ. We derived three general properties of $P_n$ from qKZ, which appeared to be extremely useful. From our experience of the direct calculation of $P_n$ with $n\le 4$ we conjectured a general ansatz for $P_n$ (\ref{Pninhom}). We established that the ansatz (\ref{Pninhom}) with the first and the second property from the Section 2 [ namely (\ref{vanish})] completely fix the answer. The third property (\ref{norm}) turns out to be a corollary. This observation allowed us to evaluate $P_5$ and $P_6$ very efficiently [ in the inhomogeneous case]. The homogeneous limit of $P_5$ reproduced the expression, which we obtained in paper \cite{BKNS} by going through hard and long computations. That time we were not sure that we will be able to calculate $P_6$ at all. Now it is possible to do it very quickly just by taking the homogeneous limit of our result in the inhomogeneous case. In the next publications we are planning to prove our general ansatz (\ref{Pninhom}) and evaluate $P(n)$ for arbitrary $n$. Actually we think that qKZ approach is so powerful that we will be able to evaluate any correlation function in the Heisenberg XXX model and to show that it has a structure similar to $P(n)$ \be F(z_1,\ldots,z_n)_{\e_1,\ldots,\e_n}\;=\;\sum_{l=0}^{[{n\over 2}]} \bigl\{ A_{n,l}^{\e_1,\ldots,\e_n} \prod_{j=1}^lG(z_{2j-1}-z_{2j}) + \mbox{permutations}\bigr\} \label{correl} \ee where $\e_j=\pm 1$ \be F(z_1,\ldots,z_n)_{\e_1,\ldots,\e_n}\;=\; \langle {\rm GS} | \prod^n_{j=1} P_j^{\e_j} | {\rm GS} \rangle, \ee and $$ P_j^{\pm}={1\pm \s^z_j\over 2} $$ We may also expect that similar to the functions $A_{n,l}=A_{n,l}^{+,\ldots,+}$ from (\ref{Pninhom}) the functions $A_{n,l}^{\e_1,\ldots,\e_n}$ are rational of their arguments $z_1,\ldots,z_n$ also. \section{Acknowledgements} The authors would like to thank A. Abanov, R. Flume, M. Jimbo, D. Kreimer, M. Lashkevich, S. Lukyanov, T. Miwa, Yu. Nesterenko, P. Pyatov, V. Tarasov, D. Zagier and A. Zamolodchikov for useful discussions. This research has been supported by the following grants: NSF grant PHY-9988566, the Russian Foundation of Basic Research under grant \# 01--01--00201, by INTAS under grants \#00-00055 and \# 00-00561. HEB would like to thank the administration of the Max-Planck Institute for Mathematics for hospitality and perfect conditions for the work. HEB and FAS are grateful to organizers of the International Workshop {\it ``Conformal Field Theory and Integrable Models''}, Chernogolovka, September 2002, Russia for opportunity to present this work. \begin{thebibliography}{99} \bibitem{B} H.~Bethe, Zeitschrift f{\"u}r Physik, {\bf 76}, 205 (1931) \bibitem{H} L.~Hulth\'{e}n, Ark. Mat. Astron. Fysik {\bf A 26}, 1 (1939). \bibitem{KBI} N. Bogoliubov, A.Izergin, V.Korepin {\it Quantum Inverse Scattering Method and Correlation Functions.} Cambridge University Press (1993) \bibitem{JMN} M. Jimbo, K. Miki, T. Miwa, A. Nakayashiki, {\it Phys. Lett.} A168 (1992) 256-263. \bibitem{JM} M. Jimbo, T. 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