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\def\lam{\frac{2\Lambda}{\pi}}
\def\xib{\bar{\xi}}
\def\a{\alpha}
\def\al{\widehat{\alpha}}
\def\n{\frac{\al}{2\pi}}
\def\x{\widehat{x}}
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\def\r#1{(\ref{#1})}
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\def\PRL#1#2#3{{\sl Phys. Rev. Lett.} {\bf#1} (#2) #3}
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\begin{document}
\begin{titlepage}
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%%March 17, 1996
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\vspace*{\fill}
\begin{center}
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%% Title:
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{\large\bf Painlev\'e V Differential Equation\\[1cm]
Describes Quantum Correlation Function of\\[1cm]
the XXZ Antiferromagnet outside of free-fermion point}
\vfill
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%%
%% Authors:
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{\sc V.E.Korepin$^1$,A.R. Its$^2$ F.H.L.\,E\ss{}ler$^3$, H. Frahm$^4$ }\\[2em]
%%
%% Addresses:
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$^1${\sl Institute for Theoretical Physics, State University of New
York at Stony Brook, Stony Brook, NY 11794-3840, U. S. A.\\[8pt]
korepin@insti.physics.sunysb.edu}\\[8pt]
$^1${\sl Sankt Petersburg Department of Mathematical Insitute of Academy of
Sciences of Russia}\\[8pt]
$^1${\sl Yukawa Institute for Theoretical Physics, Kyoto University, Japan}\\[8pt]
$^2${\sl Department of Mathematical Sciences,
Indiana University-Purdue University at Indianapolis (IUPUI),
Indianapolis, IN 46202--3216, U. S. A.}\\[8pt]
$^3${\sl Department of Physics, Theoretical Physics, 1 Keble Road,\\
\noindent Oxford OX1 3NP, Great Britain}\\[8pt]
$^4${\sl Institut f\"ur Theoretische Physik, Universit\"at Hannover,
D-30167~Hannover, Germany}\\[8pt]
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vfill
ABSTRACT
\end{center}
\begin{quote}
We consider quantum correlation functions of the antiferromagnetic
spin-$\frac{1}{2}$ Heisenberg XXZ spin chain in a magnetic field. We
show that for a magnetic field close to the critical field
correlation functions can be expressed in terms of the solution of the
Painlev\'e V transcendent. This establishes a relation between
solutions of Painlev\'e V differential equations and quantum correlation
functions in models of {\sl interacting} fermions. Painlev\'e
transcendents were known to describe correlation functions in models
with free fermionic spectra.
\end{quote}
\vfill
PACS-numbers:
75.10.Jm %%Quantized spin models
02.30.Jr %%Partial Differential equations
\setcounter{footnote}{0}
\end{titlepage}
%%
%%
%%
\section{Introduction}
%%
%%
%%
In this letter we continue our investigation of zero-temperature
correlation functions of the XXZ Heisenberg model in the
critical regime $-1<\Delta<1$ in an external magnetic field.
The XXZ hamiltonian is given by
\begin{equation}
{\cal H} = \sum_{j} \sigma_j^x \sigma_{j+1}^x
+ \sigma_j^y \sigma_{j+1}^y
+ \Delta\ (\sigma_j^z \sigma_{j+1}^z-1)
- h \sigma_j^z\ ,
\label{xxz}
\end{equation}
where the sum is over all integers $j$, $L$ is the length of the
lattice, $\sigma^\alpha$ are Pauli matrices
and $h$ is an external magnetic field. For later convenience we define
$\Delta = \cos(2\eta)$, where ${\pi\over 2}<\eta< \pi$. The free
fermionic point in this notation is $\eta=\frac{3\pi}{4}$. The model
\r{xxz} can be solved by means of the Bethe Ansatz, which yields a
description of the spectrum and eigenstates (with $N$ down spins and
$L-N$ up spins) in terms of the roots of the following set of coupled
algebraic equations \cite{orbach,yaya}
\be
\left(\frac{\sinh(\l_j-i\eta)}{\sinh(\l_j+i\eta)}\right)^L=-\prod_{k=1}^N
\frac{\sinh(\l_k-\l_j+2i\eta)}{\sinh(\l_k-\l_j-2i\eta)}\ ,
j=1\ldots N\ .
\label{bae}
\ee
For the case $\Delta > -1$ it was proved by C.N. Yang and C.P. Yang in
\cite{yaya} that the ground state is characterized by a set of
{\sl real} $\lambda_j$ subject to the equations \r{bae}.
In the thermodynamic limit the ground state is described by means of an
integral equation for the density of spectral parameters $\rho(\lambda)$
\bea
2\pi \rho(\lambda) - \int_{-\Lambda}^{\Lambda} d\mu\
K(\lambda ,\mu)\ \rho(\mu)\ &=&
\frac{-\sin(2\eta)}{\sinh(\lambda -i\eta) \sinh(\lambda +i\eta)}\nn
K (\m, \l ) &=& \frac{\sin(4\eta)}{\sinh(\mu- \lambda
+2i\eta)\sinh(\mu - \lambda -2i\eta)}\ .
\label{gsie}
\eea
Here the integration boundary $\La$ is a function of the magnetic
field $h$. The physical picture of the ground state is that of a
filled Fermi sea with boundaries $\pm \Lambda$. The dressed energy of
a particle in the sea is given by the solution of the integral
equation
\be
\epsilon(\lambda) - {1\over 2\pi}\int_{-\Lambda}^{\Lambda} d\mu\
K(\lambda ,\mu)\ \epsilon(\mu)\ = 2h -{2 (\sin(2\eta))^2\over
\sinh(\lambda -i\eta) \sinh(\lambda +i\eta) }\ .
\ee
The condition that the dressed energy vanishes at the Fermi edge
$\epsilon(\pm\La) = 0$ determines $\La$ as a function of the magnetic
field $h$.
It was shown in \cite{yaya} that the ground state of \r{xxz}
for $|\Delta|<1$ is partially magnetized for magnetic fields
$hh_c$ the ground
state is the saturated ferromagnetic state. For $h\to 0$ the integration
boundary $\La$ tends to $\infty$, whereas for $h\to h_c$ $\La\to 0$.
Below we only consider the region $h2$.
Using \r{szsz} we are now in a position to determine the
large-distance asymptotics of the $\langle\sigma^z_m\sigma^z_1\rangle$
correlation functions. By acting with (twice) the lattice laplacian on
\r{qs} we obtain
\be
-\frac{2}{m^2\pi^2}\left(1+\frac{8\La\cot(2\eta)}{\pi}+{\cal
O}(\La^2)\right).
\ee
This does not yet include the contribtution from the $\cos$-term in
\r{res}, which is very small as far as $\langle (Q_1(m))^2\rangle$ is
concerned but becomes important upon differentiation. The leading
contribution can be obtained by the methods explained in the appendix
and is found to be
\be
\left(\frac{2}{\pi^2}+{\cal O}(\La)\right)\cos\left(4\La m|\cot\eta|(
1+{\cal O}(\La))\right)\frac{1}{m^\theta}\ ,
\ee
where
\be
\theta = 2+\frac{8\La\cot(2\eta)}{\pi}+{\cal O}(\La^2).
\ee
Our final result for the first three terms of asymptotics of the
correlation function is
thus
\bea
\langle\sigma^z_m\sigma^z_1\rangle &=&
1-\frac{8\La|\cot(\eta)|}{\pi}
-\frac{2}{m^2\pi^2}\left(1+\frac{8\La\cot(2\eta)}{\pi}\right)
+\frac{2}{\pi^2m^\theta}\cos(4\La m|\cot\eta|)+\ldots\ ,\nn
\eea
where the errors in the $\La$-expansion are given above and where we
also have not taken into account subleading terms in $m$. This agrees
with the result obtained by means of finite-size corrections and
Conformal Field Theory.
%%Similarly, one obtains for the short
%%distance behaviour of the Fredholm
%%determinant
%%\bea
%% \ln(\det(1+\widehat{V}_0)) &=&
%% -{2\gamma\over\pi}\Lambda\x
%% - {2\gamma^2\over\pi^2}(\Lambda\x)^2\ ,\quad y\to 0\ , \nn
%% \gamma &=& 1- \exp(\al)\ .
%%\eea
%%The expectation value can be evaluated here as well with the result
%%\bea
%%\langle e^{\a Q_1(m)}\rangle &=& 1+\lam\cot(2\eta)
%%(1+e^\a)+{\cal O}(\La^2)\nn
%%&&+m|\cot\eta|\left(\frac{2\La}{\pi}(e^\a-1)
%%+\frac{4\La^2}{\pi^2}\cot(2\eta)(e^\a-1)+{\cal O}(\La^3)\right)\nn
%%&&-m^2\left(3\cot^2\eta\left(\lam\right)^3
%%\cot(2\eta)e^\a(e^\a-1)^2+{\cal O}(\La^4)\right).\nn
%%\eea
\section{Summary and Conclusion}
In this letter we have established a connection between the generating
functional of correlation functions $G(m)$ \r{gm} (and thus the
correlator $\langle \sigma^z_m\sigma^z_1\rangle$) of the spin
$\frac{1}{2}$ Heisenberg XXZ model in a magnetic field close to
critical and the Painlev\'e V differential equation. Painlev\'e
transcendents were known to describe correlation functions for models
with free-fermionic spectra \cite{pain,barry}. For generic magnetic fields
in the XXZ spin chain the general approach of \cite{vladb} should be
followed: the determinant representation \r{final}, \r{vcont} should
be used to embed the quantum correlation function in an integrable
system of integro-difference equations and one then should solve the
associated Riemann-Hilbert problem.
\section*{Acknowledgements}
We are grateful to Barry McCoy for helpful discussions and to Gordon
Chalmers for consultations on normal ordering.
This work was partially supported by the Deutsche
For\-schungs\-gemein\-schaft under Grant No.\ Fr~737/2--1 and by the
National Science Foundation (NSF) under Grants No.\ PHY-9321165 and
No.\ DMS-9501559.
F.H.L.E.\ is supported by the EU under Human Capital and Mobility
fellowship grant ERBCHBGCT940709.
\appendix
\section{Appendix}
In this appendix we discuss how to evaluate the expectation value
with respect to the dual quantum fields and explicitly evaluate the
quantities $\langle(Q_1(m))^k\rangle$. According to \r{simpl}
\be
\langle (Q_1(m))^k\rangle =
\nddv\frac{\partial^k}{\partial\a^k}\bigg|_{\a=0}
\det{\left(1+\widehat{V_0}\right)}\dv
{\left(1+\frac{2\La}{\pi}\cot(2\eta)+{\cal O}(\La^2)\right)}\ .
\label{app1}
\ee
Using \r{res} for the asymptotics of the logarithm of the determinant
and expanding according to \r{explog} we see that in order to get the
leading asymptotics we need to evaluate the derivatives of expectation
values of the form
\be
\nddv e^{\frac{2}{\pi}\La\x\al}e^{\frac{\al^2}{2\pi^2}\ln(4\La y)}
(g(\n))^2\left[1+\frac{\al^2}{2\pi^2}\frac{\x_p+\x_q}{y}\right]
\left[1+ \frac{\al^3}{8\pi^3\La y}\right]\dv\ .
\label{app2}
\ee
Using the commutation relations \r{xa} we see that we can bring all
terms into the form (we use that $\nddv \x_q =0$)
\be
\nddv \x_p^m e^{\frac{2}{\pi}\La \x\al} F(\al)\dv\ ,
\label{appp}
\ee
where $F(\al)$ is only a function of $\al$ and contains no $\x_p$'s or
$\x_q$'s. The central identities we will use in order to evaluate the
expectation values are
\bea
z_m=\nddv \x_p^me^{\frac{2}{\pi}\La \x\al} F(\al)\dv &=&
\frac{1}{\kappa}\nddv
\left(\frac{\x_p}{\kappa}+\frac{2\La\cot(2\eta)}{\pi\kappa}y\right)^m
e^{\frac{2\La\a}{\pi\kappa}(y+\x_p)}F(\al)\dv.\nn
\label{a1}
\eea
where $\kappa = {1-\lam\cot(2\eta)}$.
The identities are established via induction. The induction start
$m=0$ is proved as follows. Expanding the exponential and using that
$\nddv \x_q =0$, $\al_p\dv=0$ and $[\al_p,\al_q]=0$ (to move all
$\al_p$'s to the right) we obtain
\be
z_0=\nddv \sum_{n=0}^\infty\frac{1}{n!}\left(\lam\right)^n
(y+\x_p)^n (\a+\al_q)^n F(\al)\dv\ .
\ee
Expanding
\be
\nddv (y+\x_p)^n (\a+\al_q)^n = \sum_{k=0}^n
\frac{n!}{k!(n-k)!} \nddv (y+\x_p)^n \al_q^k \a^{n-k}
\ee
and then using the commutation relations
\be
\nddv [f(\x_p),\al_q^k] =\nddv (\cot(2\eta))^k f^{(k)}(\x_p)\ ,
\ee
where $f^{(k)}$ is the k'th derivative of the function $f$,
we arrive at
\bea
z_0&=&\sum_{n=0}^\infty\sum_{k=0}^n \left(\lam\right)^n
\frac{n!(\cot(2\eta))^k\a^{n-k}}{k![(n-k)!]^2} \nddv (y+\x_p)^{n-k}
F(\al)\dv\nn
\eea
We now use the integral representation
\be
\frac{1}{(n-k)!} = \frac{1}{2\pi i}\oint dt\ \frac{e^t}{t^{n-k+1}}\ ,
\ee
where the integration contour is a small circle around the origin (and
we integrate in the mathematically positive direction) in order to be
able to perform the $k$-summation (which is of the form of a
binomial sum)
\bea
z_0&=&\frac{1}{2\pi i}\oint dt \frac{e^t}{t}
\sum_{n=0}^\infty\left(\lam\right)^n
\nddv [\cot(2\eta)+\frac{\a}{t}(y+\x_p)]^n F(\al)\dv\nn
\label{intermed}
\eea
The $n$-summation can be performed using $(1-z)^{-1} =
\sum_{k=0}^\infty z^{k}$. Finally we perform the $t$-integration
formally using the identity
\be
\frac{1}{2\pi i}\oint dt\ \frac{e^t}{t-O(\a)} = e^{O(\a)}\ ,
\ee
where $O(\a)$ is an operator depending on $\a$. Here we need to keep
in mind that we are interested in evaluating derivatives with respect
to $\a$ at $\a=0$. This yields the result \r{a1} for $m=0$.
The induction step goes as follows. We rewrite $z_m$ as
\be
z_m=\nddv \x_p^{m-1} [\x_p,e^{\frac{2}{\pi}\La \x\al} F(\al)]\dv\ ,
\ee
where we used that $\x_p\dv=0$. Evaluating the commutator using
\be
\x_p f(\al)\dv = \cot(2\eta)f^\prime(\al)\dv\ ,
\label{xp}
\ee
where $f^\prime$ is the derivative of $f$ and collecting terms we
obtain
\be
z_m=\frac{1}{\kappa}\nddv \x_p^{m-1}\left[
\cot(2\eta)e^{\frac{2\La}{\pi}\x\al}F^\prime(\al)+\frac{2\La}{\pi}\cot(2\eta)
ye^{\frac{2\La}{\pi}\x\al}F(\al)\right]\dv\ .
\ee
Using the induction assumption and again \r{xp} then yields the
desired result \r{a1}.
Let us now demonstrate how to evaluate the leading contributions to
\r{app2} for $\langle Q_1(m)\rangle$ and $\langle
(Q_1(m))^2\rangle$. They are given by
\bea
\langle (Q_1(m))^l\rangle\bigg|_{\rm lead} &=&
\frac{\partial^l}{\partial\a^l}\bigg|_{\a=0}
\nddv e^{\frac{2}{\pi}\La\x\al}e^{\frac{\al^2}{2\pi^2}\ln(4\La y)}
(g(\n))^2\dv {\left(1+\frac{2\La}{\pi}\cot(2\eta)+{\cal
O}(\La^2)\right)}\ .\nn
\eea
We apply \r{a1} with $m=0$ and $F(\a) =
e^{\frac{\al^2}{2\pi^2}\ln(4\La y)}(g(\n))^2$, then perform the
differentiations with respect to $\a$ and set $\a$ to zero, and
finally use \r{xp} to evaluate the expectation value using the fact
that $\nddv \al_q =0$, $\al_p\dv =0$. The function $g$ has the
properties that $g(0) =1$, $g^\prime(0)=0$ and $g^{\prime\prime}(0)
=2(1+\gamma)$, where $\gamma$ is Euler's constant, which leads to the
result
\bea
\langle Q_1(m)\rangle\bigg|_{\rm lead} &=&\lam m|\cot\eta|
{\left(1+\frac{6\La}{\pi}\cot(2\eta)+{\cal
O}(\La^2)\right)}\nn
\langle (Q_1(m))^2\rangle\bigg|_{\rm lead} &=&
\left((\lam m|\cot\eta| )^2+\frac{\ln(4\La m|\cot\eta|
)+1+\gamma}{\pi^2}\right)\times\nn
&&\times(1+\frac{8\La\cot(2\eta)}{\pi} +{\cal O}(\La^2)).
\eea
The contributions of the subleading terms can be taken into account in
an analogous way, which leads to the results \r{qs}. We note that
contributions from further subleading terms are of higher order in
$\La$ and $y$.
%%\setlength{\baselineskip}{13pt}
\setlength{\baselineskip}{18pt}
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\end{document}