Finite temperature correlation function for one-dimensional Quantum Ising model: the virial expansion.
S. A. Reyes and A. M. Tsvelik$^\dagger$
We rewrite the exact expression for the finite temperature two-point correlation function for the magnetization as a partition function of some field theory. This removes singularities and provides a convenient form to develop a virial expansion (the expansion in powers of soliton density).
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\begin{document}
%\draft
\title{Finite temperature correlation function for one-dimensional Quantum Ising model: the virial expansion. }
\author{S. A. Reyes and A. M. Tsvelik$^\dagger$}
\affiliation{ Department of Physics and Astronomy, SUNY at Stony Brook, Stony Brook, NY 11790-3800, USA;\\ 
 $^\dagger$Department of  Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY 11973-5000, USA}
\date{\today}


\begin{abstract}
 We rewrite the exact expression for the  finite temperature two-point correlation function for the magnetization as a partition function of some field theory. This removes singularities and provides a convenient form to develop a virial expansion (the expansion in powers of soliton density). 





\end{abstract}

\pacs{75.10.Jm, 75.50.Ee}
\maketitle
\narrowtext

\sloppy
 
 To calculate correlation functions in strongly correlated systems is not an easy task, even if the corresponding models happen to be integrable. 
%Despite enormous effort of the past 20 years, there are unsolved problems of the most general kind. 
For models with dynamically generated spectral gaps the most powerful technique is the formfactor approach pioneered  by Karowski {\it et. al.} \cite{karowski} and perfected by Smirnov \cite{smirnov}. This approach works wonderfully for zero temperature, but  encounters difficulties at $T \neq 0$. These difficulties are related to singularities in the operator matrix elements (formfactors). These singularities exist for operators nonlocal with respect to solitons, they originate from forward scattering processes and their treatment requires careful infrared regularization. Despite long efforts a correct regularization  has not been yet found. However, for models of free fermions (such as the XY model or the Quantum Ising model), there are alternative means to calculate the correlation functions which allow to bypass the above problems. These alternative approaches include the determinant representation of the correlation functions \cite{its},\cite{korepin} and the semiclassical method \cite{sachdev} (which may have much wider application, see \cite{rapp},\cite{modsubir}). For these results to have a greater use one has to establish their relationship with the formfactor approach. A step in this direction was made in   \cite{akt} where the semiclassical results \cite{sachdev},\cite{rapp},\cite{modsubir} were reproduced by summing up the leading singularities in the formfactor expansion.  Such summation was restricted to the leading order in the soliton density $n \sim \exp(- M/T)$ ($M$ is the spectral gap). In this paper we describe  a formfactor-based representation for correlation functions which, though in its present form is valid only for models of free fermions, is rather suggestive and may give rise to useful generalizations in the future. For the  Quantum Ising (QI) model, which is the main object of this paper, this procedure naturally gives rise to  a virial expansion of the dynamical spin susceptibility. 

  The QI model  Hamiltonian is  
\bea
 H = \sum_n[- J\s^z_n\s^z_{n+1} + h\s^x_n] \label{is}
\eea
where $s^z,s^x$ are the Pauli matrices. By the Jordan-Wigner transformation this Hamiltonian can be transformed into the Hamiltonian of non-interacting fermions:
\bea
H = \sum_p \epsilon(p)F^+_pF_p ;~~ \epsilon(p) = \sqrt{(J- h)^2 + 4Jh\sin^2(p/2)}, \label{isF}
\eea
Hamiltonian (\ref{is})  possesses a property of self-duality: the transformation $\mu^z_{n+1/2} = \prod_{j<n}\s_j^x, \mu^x_{n+1/2} = \s^z_n\s^z_{n+1}$ preserves both  the commutation relations and the form of the Hamiltonian: 
\bea
H = \sum_n[- h\mu^z_{n-1/2}\mu^z_{n+1/2} + J\mu^x_{n+1/2}]
\eea
 As follows from the form of the dispersion law (\ref{isF}), $h=J$ is the critical point. It is easy to see that at $T=0$ it separates the regions where $\la \s^z\ra \neq 0 (J >h)$ and $\la\mu^z_{n+1/2}\ra  \neq 0 (J < h)$. Operators $\s^z$ and $\mu^z$ are respectively called order and disorder parameter operators. The duality allows to study the correlation functions at one side of the transition only. For instance, a correlation function of $\s^z$ at $J > h$ coincides with the correlation function for $\mu^z$ with $J,h$ interchanged. 

 Though one cannot observe a single fermion, the fermionic statistics can be indeirectly tested by measuring the correlation functions of $\s^x$. The operator $\s^x $ (as well as $\mu^x$) is  local in fermions: 
\bea
&& \s^x(x) = \sum_k\re^{-\ri q x}\gamma(k)\gamma(k-q)\hat F_k\hat F_{q -k}, ~~ \hat F_k = 
\hat F^+_{-k}, \nonumber\\
&& \gamma(k) = \sqrt{1 + k/\epsilon(k)}.\label{edensity}
\eea
Its  finite temperature correlation functions are just polarization loops; they clearly contain the Fermi distribution functions of the fermion fields $F$. Since the order  ($\s^z$) and disorder parameter ($\mu^z$) fields are nonlocal in terms of fermions, their correlation functions are more complicated. 

 
  The duality allows us to consider $h > J$ or $h < J$ phase only. In this paper we will study the two-point correlation functions of the $\s^z$ and $\mu^z$ operators in the ``ordered'' phase $h < J$.  For technical reasons it will be convenient to work in the limit $|J -h| << J$ when the spectral gap is much smaller than the bandwidth and one can formulate a continuous description. In this limit the excitation spectrum is relativistic $\epsilon(p) = \sqrt{c^2p^2 + M^2}$, $c^2 = Jh$. Energy and momentum of a quasi-particle are conveniently parameterized by 
a rapidity, $\theta$, ($cp = M\sinh\theta$). Then the eigenstates of Hamiltonian (\ref{is}) are 
labeled  by sets of 
rapidities, $\{\theta_i\}$, such that the energy and momentum of the system are equal to
\bea
E = M\sum_{i=1}^n\cosh\theta_i, ~~ P = c^{-1}M\sum_{i=1}^n\sinh\theta_i.
\eea
Below we set $c=1$. 
  
A convenient finite temperature expression for the  two point correlation functions of $\s$ and $\mu$ was derived by Bourgij and Lisovyy\cite{kiev}. This expression for the Matsubara time correlation function is manifestly free of singularities and has the following form:   
\bea
&&\la\s(\tau,x)\s(0,0)\ra = 
CM^{1/4}\re^{-|x|\Delta(T)}\times\label{kiev1}\\
&&\sum_{N=0}^{\infty}\frac{T^{2N}}{(2N)!}
\sum_{q_1,...q_{2N}}\prod_{i=1}^{2N}
\frac{\re^{-|x|\epsilon_i - \ri \tau q_i - \eta(q_i)}}{\epsilon_i}\prod_{i > j}
\left(\frac{q_i - q_j}{\epsilon_i + \epsilon_j}\right)^2\nonumber
\eea
%\end{widetext}
where $\tau$ is imaginary time. The same  expression holds for $\mu^z$, but with $2N$ replaced by $2N +1$.  $q =2\pi Tm$ ($m$ integer), and $\epsilon(q) = \sqrt{M^2 + q^2}$. The term in the exponent is ($\beta = 1/T$) 
\bea
\eta(q) = 
\frac{2\epsilon(q)}{\pi}\int_0^{\infty}\frac{\rd x}{\epsilon^2(q) + x^2}\ln\coth[\beta\epsilon(x)/2],
\eea
and 
 \bea
\Delta(T) = \int_{-\infty}^{\infty}\frac{\rd p}{\pi}\ln\{\coth[\beta\epsilon(p)/2]\} 
\eea
The symmetry breaking transition at $T=0$ leads to a finite magnetization,  
$\la\s\ra = \pm [CM^{1/4}]^{1/2}$. This is reflected in the zeroth order term in Eq. (\ref{kiev1}).

\begin{figure}
[ht]
\begin{center}
\epsfxsize=0.2\textwidth
\epsfbox{matrix.eps}
\end{center}
\caption{A graphic representation of Eq.(\ref{truth}). The ellipsises are formfactors of $\s$ operator. Lines with left arrows are $f^{(+)}(\theta)\exp[\ri t \epsilon(\theta) + \ri x p(\theta)$, lines with the right arrows are $f^{(-)}(\theta)\exp[-\ri t \epsilon(\theta) - \ri x p(\theta)]$. }
\label{cont}
\end{figure}
  


In \cite{akt} Eq. (\ref{kiev1}) was rewritten in the form which allowed an analytic continuation for real time. The $N$-th term in the square brackets of Eq. (\ref{kiev1}) is given as \cite{remark}  
\begin{widetext}  
\bea
\sum_{n=1}^{2N}\frac{1}{n!(2N-n)!}\int\prod_{i=1}^n\frac{\rd\theta_i}{2\pi} 
f^{(+)}(\theta_i)\re^{\tau\epsilon_i + 
\ri x p_i}\prod_{j=1}^{2N-n}\frac{\rd\theta'_j}{2\pi} f^{(-)}(\theta_j)\re^{ -\tau\epsilon_j - \ri x p_j}\frac{\prod_{i>k}\tanh^2[(\theta_i - \theta_k)/2]\prod_{j> p}\tanh^2[(\theta'_i - \theta'_p)/2]}{\prod_{i,p}
\tanh^2[(\theta_i - \theta'_p + \ri 0)/2]} \label{truth}
\eea
\end{widetext} 
where 
\bea
&& f^{(+)}(\theta_i) = \frac{\re^{\eta^{(+)}(\theta)}}{[\re^{\beta\epsilon(\theta)} -1]},  f^{(-)}(\theta_i) = \frac{\re^{-\eta^{(-)}(\theta)}}{[1 - \re^{-\beta\epsilon(\theta)}]}\label{eta}\\
&& \eta^{(\pm)}(\theta) = \frac{\ri M\sinh\theta}{\pi}\int_{-\infty}^{\infty}
\frac{\rd x \ln\{\coth[\beta\epsilon(x)/2]\}}{x^2 - M^2\sinh^2(\theta \pm \ri 0)} \nonumber
\eea
 and $p = M\sinh\theta, \epsilon = M\cosh\theta$. In Eq.(\ref{truth}), 
$n$ and $N-n$ are numbers of particles and antiparticles. Now we  rearrange the double sum (\ref{truth}) in such a way that we first sum all terms which contain a fixed difference between numbers of particles and antiparticles $2N -2n = 2k$. Then such term  in  Eq.(\ref{truth})  can  be represented as an integral of the correlation function of a Gaussian field theory:
\begin{widetext}
\bea
\frac{a_0^{-4k^2}}{(N-k)!(N+k)!}\int\prod_{i=1}^{N-k}\frac{\rd\theta_i}{2\pi a_0} 
f^{(+)}(\theta_i)\re^{\tau\epsilon_i + 
\ri x p_i}\prod_{j=1}^{N +k}\frac{\rd\theta'_j}{2\pi a_0} f^{(-)}(\theta_j)\re^{ -\tau\epsilon_j - \ri x p_j}\la \prod_i^{N - k}\re^{\ri\Phi(\theta_i + \ri a)}\prod_j^{N + k}\re^{-\ri\Phi(\theta_j' - \ri a)}\re^{2k\ri\Phi(\infty)}\ra_0 \label{corr}
\eea
\end{widetext}
where 
\bea
\la\Phi(\theta_1)\Phi(\theta_2)\ra \equiv G_0(\theta_{12}) = -\ln\left[\tanh^2(\theta_{12}) + a_0^2\right]
\eea
and $a >> a_0 \rightarrow 0$. Eq.(\ref{corr}) represents one term in the perturbative expansion of the partition function of the theory with the action 
\bea
&& S= \frac{1}{2}\int \rd\theta_1\rd\theta_2\Phi(\theta_1)G_0^{-1}(\theta_{12})\Phi(\theta_2) + \nonumber\\
&& \int \frac{\rd\theta}{2\pi a_0} V[\theta,\Phi(\theta)]\label{action}\\
&& V = f^{(+)}(\theta)\re^{ \tau M\cosh\theta +\ri x M\sinh\theta}\re^{\ri\Phi(\theta + \ri a)} + \nonumber\\
&& + f^{(-)}(\theta)\re^{ -\tau M\cosh\theta - \ri x M\sinh\theta}\re^{-\ri\Phi(\theta -\ri a)}
\eea
In this theory $x,t$ are external parameters. The field $\Phi$ lives on an infinite line in $\theta$ space.
It is easy to see that the entire correlation function can be written as 
\bea
&&\la\s(\tau,x)\s(0,0)\ra = \\
&& CM^{1/4}\re^{-|x|\Delta(T)}Z_0(x,t)
\mbox{lim}_{R \rightarrow \infty}\sum_{k=-\infty}^{\infty}\frac{\la\re^{2\ri k\Phi(R)}\ra_V}{a_0^{4k^2}}\nonumber\\
&& Z_0 = \int D\Phi \re^{-S[\Phi]}/\int D\Phi \re^{-S_0[\Phi]}\nonumber
\eea
and $\la...\ra_V$ stands for averaging with action (\ref{action}).  Already this form leads to significant simplifications. Namely, the partition function can be written as an exponent of the free energy $Z_0 = \exp[-F(x,t)]$. The latter one is represented as a sum of the cumulants:
\bea
&& F = -\sum_{N=1}^{\infty}\frac{1}{(N!)^2}\int\prod_{i=1}^{N}\frac{\rd\theta_i}{2\pi a_0} 
f^{(+)}(\theta_i)\re^{\tau\epsilon_i + 
\ri x p_i}\times\label{F}\\
&& \prod_{j=1}^{N}\frac{\rd\theta'_j}{2\pi a_0} f^{(-)}(\theta_j)\re^{ -\tau\epsilon_j - \ri x p_j}\la\la \prod_i^{N}\re^{\ri\Phi(\theta_i + \ri a)}\prod_j^N\re^{-\ri\Phi(\theta_j' - \ri a)}\ra\ra_0 \nonumber
\eea
where the subscript 0 stands for averaging with $V =0$ and
\begin{widetext} 
\bea
&&\la\re^{2\ri k\Phi(R)}\ra_V   = \sum_{N=|k|}^{\infty}\frac{1}{(N -k)!(N + k)!}\int\prod_{i=1}^{N+k}\frac{\rd\theta_i}{2\pi a_0} 
f^{(+)}(\theta_i)\re^{\tau\epsilon_i + 
\ri x p_i}\prod_{j=1}^{N -k}\frac{\rd\theta'_j}{2\pi a_0} f^{(-)}(\theta_j)\re^{ -\tau\epsilon_j - \ri x p_j}\times\nonumber\\
&&\la\la \prod_i^{N +k}\re^{\ri\Phi(\theta_i + \ri a)}\prod_j^N\re^{-\ri\Phi(\theta_j' -\ri a)}\re^{2\ri k\Phi(R)}\ra\ra_0 \label{corr1}
\eea
\end{widetext}
%In the limit $T << M$ these  expansions contain a small parameter $\re^{-\beta M}$. 


 All transformations so far have been exact. Now we would like to concentrate on the causal  Green's functions. To obtain them one has to replace $\tau$ with $\ri t$ in (\ref{truth}). Assuming that $T \ll M$, we consider the region of frequencies  $|\omega| < 2M$, where  the  only terms of the expansion (\ref{truth}) contributing to the spectral function are those which contain {\it equal} number of particles and antiparticles. As we shall demonstrate, all formfactor singularities are contained in the first term  of expansion (\ref{F}). Since $|f^{(+)}| \sim \exp(-\beta M) << 1, |f^{(-)}| \sim 1$, this expansion  is in powers of soliton density $\exp(-\beta M)$. The first term is given by 


%\begin{figure}
%[ht]
%\begin{center}
%\epsfxsize=0.2\textwidth
%\epsfbox{factor.eps}
%\end{center}
%\caption{In the approximation described in the text the diagram of Fig. 1 with %$n =m$ factorizes into a product  of independent integrals. }
%\label{cont}
%\end{figure}
\begin{widetext}   
\begin{eqnarray}
&& -F^{(1)}(t,x) = \frac{1}{4\pi^2}\int \rd\theta_1\rd\theta_2 f^{(+)}(\theta_1)f^{(-)}(\theta_2)
\frac{\exp\left(\ri \{ t[\epsilon(\theta_1)- \epsilon(\theta_2)] + x[p(\theta_1)- p(\theta_2)]\}\right)}
{\tanh^2(\theta_{12} + \ri 0)/2} \nonumber\\
&& \approx \frac{1}{2\pi^2}\int \rd\theta\rd v f^{(+)}(\theta + v)f^{(-)}(\theta -v)
\frac{\exp\left[\ri  vM(t\sinh\theta +x\cosh\theta)\right]}
{(v + \ri 0)^2}\label{R}\\
&&\approx  \theta(t - |x|)\left\{\frac{1}{\pi}\int _{\tanh\theta < - |x|/t}\rd\theta f^{(+)}(\theta)f^{(-)}(\theta)M\left[(t\sinh\theta + x\cosh\theta)|\right] + \frac{2\ri}{\pi}g[\theta = \tanh^{-1}(x/t)]\right\}
\nonumber
\end{eqnarray}
\end{widetext}
where $
g(\theta) = f^{(+)}(\theta) - f^{(-)}(\theta)$ 
and, as follows from  
 (\ref{eta}) $
f^{(+)}(\theta)f^{(-)}(\theta) = \{4\cosh^2[\beta\epsilon(\theta)/2]\}^{-1}$.% \label{exactd}
%\eea
 Here  we assumed that $Mt,(Tt)^{1/2}\gg [1-(x/t)^2]^{-1/2}$.
 The higher order cumulants contain higher powers of $\exp(-\beta M)$ and  also do not contain positive powers of $t$. This justifies keeping the exact distribution function in the real part of (\ref{R}). This cannot be done for the imaginary part, since  the second cumulant  gives a time independent contribution $\sim \exp(-2\beta M)$.  Within these limits  we obtain the following results:
\begin{widetext}
\bea
&&\la\s(x,t)\s(0,0)\ra_T  = 
 CM^{1/4}\theta(t)\exp(-\delta\Delta |x|)\exp\left\{- \frac{1}{4\pi}\int \rd p\frac{|tv(p) - x|}{\cosh^2[\beta\epsilon(p)/2]} - \frac{4\ri}{\pi}\exp[- \beta M/\sqrt{1 - (x/t)^2}]
%n_F\left(\epsilon = \frac{M}{\sqrt{1 - (x/t)^2}}
\right\}, t > |x| \nonumber\\
&& \la\s(x,t)\s(0,0)\ra_T  = 
 CM^{1/4}\theta(t)\exp(-\Delta |x|), ~~ |x| > t \label{R1}\\
&&\la\mu(x,t)\mu(0,0)\ra_T  =\la\mu(x,t)\mu(0,0)\ra_{T=0}\la\s(x,t)\s(0,0)\ra_T \label{R2}\\
&& \delta\Delta = \frac{1}{\pi}\int \rd p\left[\ln\coth(\beta\epsilon/2) - \frac{1}{2\cosh^2(\beta\epsilon/2)}\right] \sim \exp(-3\beta M)\nonumber
\eea
\end{widetext}
 The imaginary part  of (\ref{R1})  in the time-like domain $t > |x|$ reflects a quantum nature of the excitations. For $T=0$ such imaginary part was first found in \cite{mccoy}. In the leading order in $\exp(-\beta M)$ Eq.(\ref{R2}) coincides with the one found in \cite{sachdev}.  At $x =0$ we have
\bea 
&& \la\s(x=0,t > 0)\s(0,0)\ra  \approx \label{asym}\\
&& CM^{1/4}\exp\left[- t/\tau_0 + \frac{4\ri}{\pi}\re^{-\beta M} + O\left(\re^{-2\beta M}\right)\right] \nonumber
\eea
where $\tau_0^{-1} = \frac{2T}{\pi}\frac{1}{\re^{\beta M} + 1}$. 
Since single solitons are not  directly observable, it is rather interesting to note that $\tau_0$ contains the distribution function of a single soliton. It would be very interesting to see what happens in correlation functions in models  containing particles with fractional statistics.  


Since our approach shares certain common features  with the Fredholm determinant representation introduced by Korepin {\it et. al.} \cite{its},\cite{korepin}, \cite{comm},\cite{oota}, we feel obliged  to comment on the subject. The main difference is that  we do not represent the correlation functions as determinants though in certain limits this is possible. For instance, if one adopts  the nonrelativistic limit $\theta << 1$ in action (\ref{action}), it can be fermionized and rewritten as a theory of free fermions. Then by integrating  over fermions one  obtains the determinant representation. However, we would like to point out that such representation is not a goal in itself. By representing correlation functions as partition functions of some field theory one already achieves a lot since now one  can concentrate on connected diagrams where it is easier to keep track of singularities. It is possible that acting  along the lines of \cite{luk} one can obtain such representations  for interacting models. 

We also would like to warn against the direct comparison of Eq.(\ref{R1}) with  a similar equation for  the $\la \s^-\s^+\ra$ correlation function in the XY model obtained in \cite{its}. This warning  is necessary because the XY model in magnetic field is rather similar to the QI model; the similarity increases when the magnetic field exceeds the band width so that the ground state becomes ferromagnetic. However, as was explained to us by Korepin (private communication), the formulae of \cite{its} were obtained for the case of weak magnetic field the XY model spectrum is a gapless, which explains the difference in the final results. 


This research was supported  by 
the DOE under contract number DE-AC02 -98 CH 10886.  We acknowledge valuable discussions with F. H. L. Essler, R. M. Konik, F. A. Smirnov, V. E. Korepin  and V. A. Fateev. 




 

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\end{document}


A further simplification occurs if we restrict ourselves to the nonrelativistic limit replacing  $\tanh\theta \approx \theta, \cosh\theta \approx 1 + \theta^2/2$. In that case this theory is equivalent to the theory of free fermions with quadratic action:
\bea
&& S_0 = \int\rd\theta_1\rd\theta_2(R^+, L^+)_{\theta_1}\left(
\begin{array}{cc}
G_{RR}^{-1}(\theta_1,\theta_2) & v^{(+)}(\theta_1)\delta(\theta_{12})\\
v^{(-)}(\theta_1)\delta(\theta_{12}) & G_{LL}^{-1}(\theta_1,\theta_2)
\end{array}
\right)\left(
\begin{array}{c}
R\\
L
\end{array}
\right)_{\theta_2}
\nonumber\\
&& G_{RR}(\theta) = (\theta + \ri 0)^{-1}, ~~ G_{LL}(\theta) = (\theta - \ri 0)^{-1} 
\eea
where 
\bea
&& v^+(\theta)= f^{(+)}(\theta)\re^{ -\tau M\cosh\theta - \ri x M\sinh\theta},\nonumber\\
&&  
v^-(\theta) = f^{(-)}(\theta)\re^{ -\tau M\cosh\theta - \ri x M\sinh\theta}
\eea
One can check that the correlation functions of fermionic bylinears  coincide with the correlation functions of the bosonic vertex operators.





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{\it Quantum Inverse Scattering Method and Correlation Functions}, Cambridge
University Press (1993) Cambridge.
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ibid, Nucl. Phys. B {\bf 567} (2000) 554.
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K. Damle and S. Sachdev, Phys. Rev. B {\bf 57} (1996) 8307.
\bibitem{note} This is relatively easy to show for the case of the Ising model.  
Here we can directly rederive the result
of Section 3 starting directing from Eq. 1 using 
infinite volume form factors.  This arrangement of poles coincides with the form
factors yielding a finite expression.  With correlators of the sine-Gordon model, it can be demonstrated that
at least the leading infinities contained in the disconnected pieces of the form factors vanish.  
To show sub-leading pieces
vanish is more difficult as the full structure of the (highly non-trivial) sine-Gordon form 
factors are needed rather
than merely the behavior of the form factor in the immediate vicinity of its annihilation poles.  
This work will
be reported shortly by the authors together with F. H. L. Essler.
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{\bf 74} (1995) 972;
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P. Prelovsek, Phys. Rev. B {\bf 55} (1997) 11029.
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{\it Dynamical Correlations and Quantum Phase Transitions in the Quantum Potts Model}, cond-mat/0507390.
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\end{thebibliography}
\end{document}










\section{The general case: Two universality classes}

The form of Eq. (\ref{truth}) has a form intimately related to the double summed Lehmann expansion
of Eq. 1.  Apart the bosonic distribution function $f(\epsilon)$ (unimportant in the limit $T \ll M$), 
Eq. (\ref{truth}) looks for all intent purpose as those subset of terms in Eq. 1 involving 
form factors (as given by Eq. 4) of exactly
N particles or excitations.  This then 
implies that the double summed Lehmann expansion is tractable.
Or put another way, we can generalize to correlation functions involving
an operator ${\cal O}$ in other integrable models 
by
using Eq. (\ref{truth}) with the Ising model form factors replaced  
by the form factors of ${\cal O}$. 
At temperatures much smaller than the gap, we replace the distribution 
functions by Boltzmann exponents. 
The asymptotics of the correlation function  is then determined solely 
by the behavior of the form factors in the vicinity of the kinematic poles. 
As we have already pointed out, 
the existence of kinematic poles is a general consequence of the theory of form factors \cite{smirnov} 
rather than a specific feature of the Ising model.  These poles are due to annihilation
processes and occur when rapidities of a particle and an anti-particle coincide.  The behavior of 
form factors in the vicinity of the poles divide all integrable models into two universality classes. 
Accordingly the resolution of the problem of singularities should not be model-specific. 

Consider a generic matrix element of operator ${\cal O}$:
\bea
 F^{\cal O}(\theta_i,a| \Tt_i\,b) 
= \la \tp_1,\ldots,\tp_n|{\cal O}|\Tt_1,\ldots,\Tt_m\ra_{ a_1,... a_n; b_1,...b_m}, \label{FF}
\eea
where $\tp_i$ and $\Tt_j$ are rapidities and $a_i$ and $b_j$ are isotopic indices of $in$ and $out$ 
particles. One can express  
the form factor (\ref{FF}) in terms of 
canonical matrix elements. Let $F^{\cal O}_{irr}$ be the irreducible part of $F^{\cal O}$ 
given by the connected diagrams. Only this part is necessary for our calculations done to 
lowest order in $\exp(- M\beta)$. 
According to Refs. \cite{balog,mussardo,konik1}, 
$F^{\cal O}_{irr}$ is equal to the matrix element 
between the vacuum and a certain excited state of $n+m$ particles:  
\bea
\label{poles}
 F^{\cal O}(\tp_i,a| \Tt_i, b)_{irr} = \la 0|{\cal O}|\Tt_1,\ldots,\Tt_m;
-\ri\pi + \tp_n \pm \ri 0,\cdots , -\ri\pi + \tp_1 \pm \ri 0 \ra_{b_1,...b_m; \bar a_n,...\bar a_1;}
\eea
where $\bar a$ indices are obtained from $a$ ones by charge conjugation.  
The choice of whether the small imaginary part, $\ri 0$, is positive or negative
determines the exact form of the disconnected parts (here ignored) of the form factor in Eq. (\ref{FF}).

\begin{figure}
%[ht]
\begin{center}
\epsfxsize=0.6\textwidth
\epsfbox{identity.eps}
\end{center}
\caption{A graphical representation of Eq. (\ref{gen}) 
written in terms of the irreducible form factors in Eq. (\ref{FF}) (the rectangles). 
The up arrows represent rapidities $\Tt_i$, while
the down ones correspond to the rapidities $-\ri\pi + \tp_j$. 
Circles at the intersections of the solid lines represent two-particle $S$ matrices.}
\label{ann}
\end{figure}

The axiom for the annihilation poles in the standard formulation is expressed in terms of the 
form factors of the type
\begin{equation}
{\cal F}^{\cal O}_{a_1,\ldots,a_n}(\theta_1,\ldots,\theta_n) 
= \la 0|{\cal O}|\theta_n,...\theta_1\ra_{a_n,\ldots,a_1}.
\end{equation}
Namely, as was shown in Ref. \cite{smirnov} 
(see also Ref. \cite{konik} in review), 
these form factors have poles at $\theta_n - \theta_j = \ri\pi$, $j<n$, 
and the residues satisfy the following relation (see Fig. \ref{ann}):
\bea
&&\ri {\rm Res}_{\theta_{n} = \theta_{n-1}+\ri\pi} 
{\cal F}^{\cal O} _{a_1,\ldots,a_n}(\theta_1,\theta_2,\ldots,\theta_n) =   
{\cal F}^{\cal O} _{a'_1,\ldots,a'_{n-2}}(\theta_1,\ldots,\theta_{n-2}) \label{gen} \\
&& \times C_{a_n,a'_{n-1}}\left[{\bf \rm I}-\re^{2\pi \ri \varphi_{a_n,{\cal O}}} 
S_{\tau_1,a_1}^{a'_{n-1},a'_1}(\theta_{n-1}-\theta_1)
\bigg( \prod_{i=2}^{n-3} S_{\tau_i,a_i}^{\tau_{i-1},a'_i}(\theta_{n-1}-\theta_i) \bigg)
S_{a_{n-1},a_{n-2}}^{\tau_{n-3},a'_{n-2}}(\theta_{n-1}-\theta_{n-2})\right] . \nonumber 
\eea 
Here $S$ is the  two-particle scattering matrix, $C$ is the charge conjugation matrix, and $\varphi_{a,{\cal O}}$
is the {\it semi-locality index} between the particle creation operator $A_a^\dagger$ and ${\cal O}$. 
We will define this index  later. 

 
To demonstrate how Eq. (\ref{gen}) can be used to extract the most singular behavior of the form factors, 
we combine it with (\ref{FF}) to obtain the following 
behavior of two- and three-particle form factors at the poles:
\bea
\la \theta, a |{\cal O}|\theta_1,b \ra &=& -\frac{
\ri(1 - \re^{2\pi\ri\varphi_{b,{\cal O}}})}{\theta_1 - \theta \pm \ri 0}\la 0|{\cal O}|0\ra \delta_{a,b} 
+ \mbox{disconnected part;}\label{two}\\
\la \theta_1, a_1;\theta_2,a_2 |{\cal O}|\theta,a \ra &=& 
-\ri\frac{\delta_{a,a_2}\delta_{a_1,a'_1} - e^{i2\pi\varphi_{a,{\cal O}}}
\delta_{a,a'_2} S^{\bar{a_2'},\bar{a_1'}}_{\bar{a_2},\bar{a_1}}(\tp_2-\tp_1)}
{\theta-\theta_2 \pm \ri 0}\la 0|{\cal O}|\tp_1-\ri\pi,\bar{a_1'}\ra \nonumber \\ 
&& -\ri\frac{\delta_{a,a'_1}S^{\bar{a_2'},\bar{a_1'}}_{\bar{a_2},\bar{a_1}}(\tp_2-\tp_1)
- e^{i2\pi\varphi_{a,{\cal O}}}\delta_{a,a_1}\delta_{a_2,a'_2}}
{\theta-\theta_1 \pm \ri 0}\la 0|{\cal O}|\tp_2-\ri\pi,\bar{a_2'}\ra \nonumber \\ 
&& + \mbox{~~disconnected part.} 
\eea
Now it is appropriate to  define  the index $\varphi$.  
We will call two operators, ${\cal O}_1$ and ${\cal O}_2$, mutually non-local if the Euclidean
correlator $\langle \ldots {\cal O}_1({\bf r}) {\cal O}_2 (0) \ldots
\rangle$ is a multi-valued function of ${\bf r}$. 
In particular, if  
under analytic continuation $z\to z \re^{2\pi \ri}$,  
$\bar z\to \bar z \re^{-2\pi \ri}$ ($z=\tau +\ri ~x$ and $\bar z=\tau -\ri ~x$), the correlator
acquires only a phase, $2\pi \varphi_{1,2}$, the
two operators are called {\em semi-local}. 
The remarkable advantage  of this 
definition is that the index $\varphi_{1,2}$ can be evaluated at small distances, 
where all correlation functions have a simple power law form.  

From Eq. (\ref{two}) it follows that an operator ${\cal O}$ with a finite vacuum expectation
has annihilation poles only if $\varphi_{a,{\cal O}}$ is not an integer, that is when  ${\cal O}$ is non-local. 
For instance,  $\s$ and $\mu$ in the Ising model are non-local with respect to the fermions 
(they acquire a phase, $\pi$, after circling around the fermions in the $\tau-x$ plane). 
However, if the operator can create an  odd number of particles, 
the annihilation poles may appear provided the $S$ matrix is non-trivial. 
For instance, the vector field ${\bf n}$ in the O(3) nonlinear sigma model belongs to the later category. 
Its form factors do have annihilation poles, although its vacuum expectation value is zero \cite{smirnov}.  

We proceed with the calculation of  the most singular parts of the form factors. We will do
it explicitly for operators with $\la 0|{\cal O}|0\ra \neq 0$. 
The maximal singularities come from the matrix elements 
which contain equal numbers of $in$ and $out$ particles. 
The contributions from all other matrix elements contain additional powers 
of either $\exp[-\beta M]$ or $\exp[- \ri tM]$. 
Since form factors with different order of rapidities are related to one 
another (Refs. \cite{smirnov,karowski,konik}), it is sufficient
to know the residues for one particular arrangement. 
For the form factors of the type in Eq. (\ref{FF}), 
we focus on the residues at  the poles at $\theta_i = \Tt_i$. 
In this case one can use the crossing symmetry of the S-matrix and obtain from Eq. (\ref{gen}) 
the relation for the residues of (\ref{FF}).  This allows the convenient graphical representation of 
Fig. \ref{ann}.  

At this point, the computation divides itself into two cases.  Each case marks out a separate universality class.
We deal with each in turn.

\subsection{Ising-like Universality Class}

In the first case, Ising-like behavior results.  
This occurs if either of following conditions hold
\vskip 10pt
1.  The $S$ matrix is diagonal.
\vskip 10pt
2.  The $S$ matrix is not diagonal, but the semi-locality index,
$\varphi_{a,{\cal O}}$ = 1/2, $a=1,\ldots, \alpha$, is independent of the particle index, $a$. 
\vskip 10pt

\noindent In these cases the residue is diagonal in the isotopic indices: $a_i = b_i$. 
For the diagonal S-matrix this is obvious. 
For the case $\varphi = 1/2$ with a non-diagonal S-matrix, some additional reasoning is required. 
Namely, one needs to apply  Eq. (\ref{gen}) in successive steps. 
For the first step we obtain the residue in the two-particle form factor. 
This yields Eq. (\ref{two}) diagonal in the isotopic indices. 
Then we use this result to calculate the coefficient at  $[(\tp_1 - \Tt_1)(\tp_2 - \Tt_2)]^{-1}$ 
in the four-particle form factor 
${\cal F}_{\cal O}(\tp_1-\ri\pi\pm\ri 0,\tp_2-\ri\pi\pm\ri 0,\Tt_2,\Tt_1)_{\bar{a_2}\bar{a_1}b_2b_1}$. 
As follows from Eq. (\ref{gen}), the S-matrices enter into the expression for this coefficient 
in the vicinity of the double pole in the combination 
\bea
S_{\bar{a}_2,\bar{a}_1}^{\bar{a}'_2,\bar{a}'_1}(\tp_{21})S_{b_2,\bar{a}'_1}^{a'_2,\bar{a}''_1}(\ri\pi + \tp_{21}) 
= S^{\bar{a}'_2,\bar{a}'_1}_{\bar{a}_2,\bar{a}_1}(\tp_{21})S^{\bar{b}_2,\bar{b}_1}_{\bar{a}'_2,\bar{a}'_1}(-\tp_{21}) 
= \delta_{a_1,b_1}\delta_{a_2,b_2}.
\eea
Here we have used the crossing symmetry and unitarity of the S-matrices. 
As we see, this residue is again diagonal in the isotopic indices and 
equal to $(-\ri)^2(1 - \re^{2\pi\ri\varphi})^2 = -4$.

Thus for these two cases, it is clear that a multiple application of formula (\ref{gen}) 
yields the main singularity for general $n$ as
\begin{eqnarray}
F^{\cal O}_{sing}(\tp_1,\ldots,\tp_n;\Tt_1,\ldots,\Tt_n) 
&=& (-\ri)^n\sum_{P_i}(1 - \re^{2\pi \ri \varphi_{a,{\cal O}}})^n
\prod_{i=1}^n(\tp_i - \Tt_{P_i} \pm \ri 0)^{-1}\la 0|{\cal O}|0 \ra;\cr\cr
&=& (-2\ri)^n\sum_{P_i}
\prod_{i=1}^n(\tp_i - \Tt_{P_i} \pm \ri 0)^{-1}\la 0|{\cal O}|0 \ra,
\end{eqnarray}
where the sum includes all permutations of $\{1,...n\}$. 
The result for the correlation function of operators, ${\cal O}$,
with {\it finite vacuum expectation values} is then essentially the same as Eq. (\ref{sachdev}) but with 
\bea
\exp[-\beta\epsilon] \rightarrow \sum_{j=1}^\alpha\sin^2(\pi\varphi_j)\exp[-\beta\epsilon_j]
= \sum_{j=1}^\alpha\exp[-\beta\epsilon_j], \label{corr}
\eea
where the sum runs over isotopic indices of the excitations.

\subsection{Non-Ising Like Universality Class}
When both the S-matrix is non-diagonal and the semi-locality index, $\phi_{a,{\cal O}}$,
is not simply $\pm 1/2$, the behavior of the correlators is considerably more complex.
Attendantly, the calculations are more difficult 
due to the fact that $\varphi_a$ depends on the particle index.  
Nevertheless the maximal residue can be calculated, although only at small rapidities 
(in the given context, that is all we need). 
We take advantage of the fact that in integrable models with a non-diagonal $S$-matrix, 
the $S$ matrix has the universal limit $S(\theta \rightarrow 0) = -P$ 
(the permutation operator). This limit  corresponds to a pure reflection. 
In general the crossover to pure reflectivity occurs at some model-dependent scale $\theta_0$. 
It may happen that this  scale is much smaller than one; then universal dynamics occurs only  
for $T \ll M\theta_0^2 < M$ when rapidities $|\tp | \ll \theta_0$ 
determine the expressions for the correlation functions. 
At these temperatures, 
one can replace all S-matrices by permutation operators which leads to 
simplifications in Eq. (\ref{gen}) (see Fig. \ref{ann1}): 

\begin{figure}
\begin{center}
\epsfxsize=0.6\textwidth
\epsfbox{iden2.eps}
\end{center}
\caption{A graphical representation of Eq. (\ref{gen}) for S=P.}
\label{ann1}
\end{figure}

As we see from Fig. \ref{ann1}, the isotopic indices of $in$ and $out$ particles must coincide. 
This allows us to simplify the notational representation of the form factor:
\[
\mbox{Res}_{\tp_i = \Tt_i} 
{\cal F}^{irr}(\tp_1,\ldots,\tp_n;a_1,...a_n|\Tt_n, ... \Tt_1; a_n,...a_1) 
\equiv (-i)^n f(a_1,...a_n)\la 0|{\cal O}|0\ra .\]
Then the equation depicted in Fig. \ref{ann1} becomes 
\bea 
f(a_1,...a_n) = f(a_2,...a_n) - \re^{\ri\phi_{a_n}}f(a_1,...a_{n-1}) \label{id}.
\eea
This recurrence relation can be solved explicitly with the result
\bea
f(a_1,...a_n) = \sum_{k=1}^n
\left[(1 - \re^{\ri\phi_k})(-1)^{(n-k)}\exp\left(\ri\sum_{j= k+1}^n\phi_j\right)\left(
\begin{array}{c}
n-1\\
k-1
\end{array}
\right)\right]\label{solution},
\eea
where $\phi_a \equiv 2\pi\varphi_{a,{\cal O}}$. 

In this case the correlator, $\la {\cal O}(x,t){\cal O}(0,0)\ra$, takes the
form
\begin{eqnarray}
\la {\cal O}(x,t){\cal O}(0,0)\ra  &=& \la {\cal O}\ra^2 \sum^\infty_{n=0} \frac{1}{n!}\sum_{a_i}|f(a_1,\ldots,a_n)|^2
\int \frac{d\tp_1}{2\pi}\cdots\frac{d\tp_n}{2\pi}\frac{d\Tt_1}{2\pi}\cdots\frac{d\Tt_n}{2\pi} \cr\cr
&& \hskip -1.5in \times
\exp\bigg[-\beta M\sum^n_{j=1}\cosh(\Tt_j)\bigg] \cr\cr
&& \hskip -1.5in \times \exp\bigg[iMt\sum^n_{j=1}(\cosh(\tp_j)-\cosh(\Tt_j))-iMx\sum^n_{j=1}(\sinh(\tp_j)-\sinh(\Tt_j))\bigg] \cr\cr
&& \hskip -1.5in \times \frac{1}{2^n}\sum^n_{k=0} C_n^k \prod_{q=1}^k\frac{1}{(\tp_q-\Tt_q-\ri 0)^2}
\prod_{p=1}^{n-k}\frac{1}{(\tp_p-\Tt_p+\ri 0)^2},\label{double}
\end{eqnarray}
where $C^k_n = ({n\atop k})$.  Again this expression arises either from using the form factors
of ${\cal O}$ in conjunction with Eq. (\ref{truth}) or directly from the double summed
Lehmann expansion (Eq. (1)).

Notice the particular arrangement of double poles in Eq. (\ref{double}).  As we 
have indicated the irreducible part of the form factor
offers some flexibility in whether to deform the poles into the upper or lower 
half plane (at the cost of changing
the disconnected pieces of the form factors).  With this particular
choice of poles, the integral can be straightforwardly evaluated  
(under the same assumption as before that the most important contribution
comes from small $\tp_i$ and $\Tt_i$). The result is 
\begin{eqnarray}\label{S}
\la {\cal O}(x,t){\cal O}(0,0)\ra  &=&  
\la {\cal O}\ra^2\sum^\infty_{n=0} \frac{(-S(x,t))^n}{2^{2n}n!} \sum_{a_i}|f(a_1,\ldots,a_n)|^2 ,
\end{eqnarray}
where
\begin{equation}
S(x,t) =  \frac{M}{\pi}\re^{-M\beta}\int d\tp \re^{-\frac{M\beta }{2}\theta^2}|x-t\tp|.
\end{equation}
This particular arrangement of poles can be justified by examining the structure of the disconnected
pieces \cite{note}.  Within the disconnected pieces lurk potential infinities (as regulated by the $i 0$ terms).  
However with our choice of poles, the infinities ultimately cancel one another.  Thus this arrangement of poles
is dictated by the requirement the form factors in their entirety (connected and disconnected pieces) 
yield a finite expression.

We now turn to the computation of the residues, $f(a_1,\ldots,a_n)$, of the poles.
To simplify the discussion, we will consider only the case where the excitation spectrum 
of the theory constitutes a doublet: $a = \s = \pm 1$. 
This situation includes many important models, in particular the sine-Gordon model. 
In that case the semi-locality index of operator ${\cal O}$ with a soliton/anti-soliton equals $\pm \phi$.  We have, for instance,  
\[
f(++) = (1-\re^{\ri\phi})^2, ~~ f(+-) = 2(1 - \re^{-\ri\phi}), ~~ f(-+) = 2(1 - \re^{\ri\phi}), ~~ f(--) = (1-\re^{-\ri\phi})^2.
\]
\noindent In this case Eq. (\ref{solution}) reduces to
\bea
\sum_{\s_i = \pm 1}|f(\s_1,...\s_n)|^2 = 2^{n+2}\sin^2(\frac{\phi}{2})\left\{C_{2n-2}^{n-1} + 2\sin^2(\frac{\phi}{2})\sum_{k > j}C_{n-1}^{k-1}C_{n-1}^{j-1}(-\cos\phi)^{k-j-1}\right\}.
\eea
To evaluate the sum, $k<j$, we rewrite it as follows:
\begin{eqnarray}
2\sum_{k > j}C_{n-1}^{k-1}C_{n-1}^{j-1}(-\cos\phi)^{k-j-1} 
&=& \frac{1}{\cos(\phi)}C_{2n-2}^{n-1}-\frac{1}{\cos\phi} 
\sum^n_{k,j}C_{n-1}^{k-1}C_{n-1}^{j-1}(-\cos\phi)^{|k-j|};\cr\cr
&\equiv& \frac{1}{\cos(\phi)}C_{2n-2}^{n-1} - \Gamma .
\end{eqnarray}
For $\cos(\phi) > 0$, we use the identities
\begin{eqnarray}
\cos (\phi )^{|k-j|} &=& \frac{1}{\pi}\int^\infty_{-\infty} \rd\omega \frac{\delta}{\omega^2+\delta^2}\re^{i\omega (k-j)};\cr\cr
\sum^\infty_{m=-\infty} \frac{\delta}{4(\omega+m\pi)^2 + \delta^2} &=& 
\frac{1}{8}\frac{\sin^2(\phi)}{\sin^2(\omega)\cos(\phi)+\sin^4(\phi/2)},
\end{eqnarray}
where $\delta = - \ln\cos\phi$, to rewrite $\Gamma$ as
\begin{eqnarray}
\Gamma = \frac{2^{2n-4}}{\pi\cos(\phi)}\int^\pi_0\rd\omega \sin^{2n-2}(\omega)\frac{\sin^2(\phi)}{\sin^2(\omega)\cos(\phi)+\sin^4(\phi/2)}.
\end{eqnarray}
If we finally use the identity to write the combinatorial factor, $C_{2n-2}^{n-1}$, in a similar integral form
\begin{eqnarray}
C_{2n-2}^{n-1} = \frac{2^{2n-2}}{\pi} \int^\pi_0 \rd\omega (\cos(\omega/2))^{2n-2},
\end{eqnarray}
we can, using Eq. (\ref{S}), represent the correlator as 
\begin{equation}\label{asymp}
\la {\cal O}(x,t){\cal O}(0,0)\ra = \frac{1}{2\pi} \int^{\pi/2}_0 \rd\omega \exp\big[-2S(x,t)\sin^2(\omega)\big]
\frac{\sin^2(\phi)}{\sin^2(\omega)\cos(\phi)+\sin^4(\phi/2)}. \label{S1}
\end{equation}
It turns out that this relation holds even if
$\cos(\phi) < 0$.  To establish this we must use instead the identity
\begin{eqnarray}
\sum^\infty_{m=-\infty} \frac{\delta}{4(\omega+m\pi)^2 + \delta^2} &=& 
\frac{1}{8}\frac{\sin^2(\phi)}{\cos^2(\omega)\cos(\phi)+\sin^4(\phi/2)}.
\end{eqnarray}
where now $\delta = -\ln |\cos(\phi )|$.  We point out that as $\phi \rightarrow \pi$, the expression
for the asymptotics of the correlator (Eq. \ref{asymp}) reproduces the result for the Ising universality class.

For $x=0$ and large $|t|$, this correlation function behaves
as
\begin{equation}
\la {\cal O}(0,t){\cal O}(0,0)\ra \sim 1/t^{1/2}.
\end{equation}
i.e. it demonstrates a comparatively slow, non-exponential, decay.
For $t=0$ and large $|x|$, we find similar behavior:
\begin{equation}
\la {\cal O}(x,0){\cal O}(0,0)\ra \sim 1/x^{1/2}.
\end{equation}
This behavior is dramatically different from the cases of a diagonal S-matrix or a non-diagonal S-matrix
with trivial non-locality phase.
In the latter two cases, we find exponential decay at large $x$ and $t$.

These results are also qualitatively 
different from the recent semi-classical analysis in Refs. \cite{kedarsubir,zarand}.
In Ref. \cite{kedarsubir} the sine-Gordon model is explicitly studied while in Ref. \cite{zarand}
the three state Potts model outside of the scaling regime is examined.
Here it appears the S-matrix
is non-diagonal \cite{note1}.  The two semi-classical analyses coincide (after a certain
identification of parameters).
But their formulae are generically at variance with ours.
While the formulae in Ref. \cite{zarand} (i.e. Eq. 12) formally coincides with our treatment at $x=0$ 
(provided one makes the identification, $1 - Q = \cos\phi$),  the 
equal time dependence however is radically different, with the semi-classical analysis indicating
that the correlation functions will decay exponentially.
We discuss this discrepancy between the semi-classical analysis
and our analysis in Section V.


\section{Low temperature charge dynamics in the sine-Gordon model}

To provide an example of a physical problem which can be solved by the method developed
in this paper, we consider the problem of charge counting statistics in the sine-Gordon model. 
The Lagrangian corresponding to this theory is 
\bea
{\cal L} = \frac{1}{16\pi}(\p_{\mu}\Phi)^2 - m^2\cos(\gamma\Phi) .
\eea
The cosine term is relevant at $\gamma < 1$. The conserved soliton charge is defined as 
\bea
Q = \frac{\gamma}{2\pi}\int \rd x\p_x\Phi .
\eea
Therefore $Q(x,t) = (\gamma/2\pi)[\Phi(x,t) - \Phi(x,0)]$ is the charge that has
passed through a point $x$ during time 
$t$. One can calculate the probability distribution 
function of $Q(x,t)$ using the results 
for the correlation function of  operators $\exp[\ri\eta\Phi]$. 
In this case $\varphi_{\pm} = \pm (\eta/\gamma)$. 

The sine-Gordon model is a good example to study as it encompasses both universality classes
as $\gamma$ is varied.  
At the
so-called reflectionless points $\gamma^2_n = 1/(n+1)$, $n>1$, 
the sine-Gordon model model has a diagonal S matrix. 
At these points the model belongs to the same universality class as the Ising model. 
Away from these values of $\gamma$, the soliton-soliton scattering is not diagonal and the model falls into the 
second universality class (although for operators $e^{\pm i\Phi\gamma/2}$ where the semi-locality index is
$\pm1/2$ 
we revert to the Ising universality class).  In this case the physics is governed by the second universality class
provided we are at a sufficiently low temperature.  
The rapidity scale governing this crossover is
$$
\theta_0 \approx \frac{\nu}{2}\cosh^{-1}\big(\cos(\frac{2\pi}{\nu})+2|\sin(\frac{\pi}{\nu})|\big),
$$
where $\nu = \gamma^2/(1-\gamma^2)$.  If $\gamma$ is close to one of the reflectionless
points, i.e. $\gamma^2 = 1/(n+1) + \delta$, the crossover rapidity simplifies
to
$$
\theta_0 \sim \frac{n+1}{n}\sqrt{\pi\delta}.
$$
The corresponding temperature
at which the crossover 
occurs is $T^* \sim \pi M \frac{(n+1)^2}{2n^2}\delta$. 

We now consider the counting statistics
present in the sine-Gordon model when it falls into each of the two universality classes.

\subsection{Counting statistics at the reflectionless points}

At $\gamma^2 = 1/(n+1)$ we are in the Ising-like universality class.
The correlation function then takes the form, as follows
from Eq. (\ref{corr}),
\bea
\la \re^{\ri\eta\Phi(x,t)}\re^{-\ri\eta\Phi(0,0)}\ra = 
C^2(\gamma,\eta)\exp\left[-2\sin^2(\pi\eta/\gamma)
\int \frac{\rd p}{\pi}\re^{-\epsilon(p)\beta}|x - t\frac{\p\epsilon(p)}{\p p}|\right]\label{sineG},
\eea
where $\epsilon (p)$ is the soliton energy and 
the factor $C(\gamma,\eta)$ is equal to 
the vacuum expectation value of $\exp[\ri\eta\Phi]$. 
This formula remains valid even for $\gamma^2 \neq (n+1)^{-1}$ provided  $\eta = \gamma/2$. The quantity $C(\gamma,\eta)$ was calculated in Ref. \cite{lukyanov}:
\bea
&&C(\gamma, \eta) = \left[\frac{a M\Gamma(1/(2-2\gamma^2))\Gamma((2-3\gamma^2)/(2-2\gamma^2))}
{4\sqrt{\pi}}\right]^{2\eta^2}\times\nonumber\\
&&\exp\left\{\int_0^{\infty}\frac{\rd y}{y}
\left[\frac{\sinh^2(2\eta\gamma y)}{2\sinh(\gamma^2 y)\sinh y\cosh[(1 - \gamma^2)y]} 
- 2\eta^2\re^{-2y}\right]\right\} .\label{factor}
\eea
Here we have assumed that the correlator, $\la \re^{\ri\eta\Phi(x,t)}\re^{-\ri\eta\Phi(0,0)}\ra$, 
satisfies the short distance normalization, 
$$
\la \re^{\ri\eta\Phi(x,t)}\re^{-\ri\eta\Phi(0,0)}\ra \sim \frac{4a^{\eta^2}}{|x-y|^{4\eta^2}},
$$
where $a^{-1}$ is the effective cutoff in the theory.
$M \sim m^{1/(1 - \gamma^2)}$ is the soliton mass. 
In the field theory limit, $M a \ll 1$, and so
the $\eta$ dependence in Eq. (\ref{factor}) is dominated by the term $(Ma)^{2\eta^2}$. 
Thus the magnitude of the propagator has a strong maximum at small $\eta$. 

We now examine the generating functional $\la \exp[\ri\omega Q(x,t)]\ra$.
This is given by 
\begin{eqnarray}
\la \re^{\ri\omega Q(x,t)}\ra &=& 
\la \re^{\ri\frac{w\gamma}{2\pi}\Phi(x,t)}\re^{\ri\frac{w\gamma}{2\pi}\Phi(x,0)} \ra \cr\cr
&=& C(\gamma, \frac{w\gamma}{2\pi})^2 \exp\bigg[-2\sin^2(\frac{\omega}{2})\int\frac{\rd p}{\pi}
\re^{-\beta\epsilon(p)}|t\partial_p\epsilon (p)|\bigg].
\end{eqnarray}
In the limit of large time the above integral finds its predominant contribution
at small $\omega$.  We then can write
\begin{eqnarray}\label{omega}
\la \re^{\ri\omega Q(x,t)}\ra &=& (\alpha aM)^{(\frac{\omega\gamma}{\pi})^2}
\exp\bigg[-\omega^2|t|/\tau_0\bigg],
\end{eqnarray}
where we have written $C(\gamma , \frac{w\gamma}{2\pi})^2 
\sim (\alpha a M)^{(\frac{\omega\gamma}{\pi})^2}$ for
some constant $\alpha$ and
where $1/\tau_0 = (T/\pi)\re^{- M/T}$.
%\bea
%\bar M = M\frac{\sqrt\pi\Gamma(1/(2-2\gamma^2))}{2\Gamma(\gamma^2/(2 - 2\gamma^2))}\exp\left\{2\int_0^{\infty}\frac{\rd y}{y}\left[\frac{y^2\gamma^2}{2\sinh(\gamma^2 y)\sinh y\cosh[(1 - \gamma^2)y]} - \re^{-2y}\right]\right\}
%\eea

The  Fourier transform of Eq. (\ref{omega}) then 
yields a Gaussian distribution function for the charge,
\begin{eqnarray}
P(\tilde{Q}) \equiv \la \delta(\tilde{Q} - Q)\ra
&=& \int \rd\omega \re^{-\ri\omega \tilde Q}\la \re^{\ri \omega Q(x,t)}\ra \cr\cr
&\sim & \s^{-1/2}\exp\left[- \tilde{Q}^2/4\s\right],
\end{eqnarray}
where
$\s = (t/\tau_0) + \frac{\gamma}{\pi^2}\ln[1/(\alpha aM)]$.


\subsection{Charge counting statistics at $\gamma^2 \neq 1/(n+1)$}
Here we consider the sine-Gordon model when it falls into the second universality class.
As we have already mentioned, at $\gamma^2 \neq 1/(n+1)$ 
there is a crossover temperature $T^*$ 
below which one can replace the S-matrix of slow solitons 
by the permutation operator.  
Then we may use Eq. (\ref{S1}), finding that
\begin{equation}
\la \re^{\ri\omega Q(x,t)}\ra = C^2(\gamma,\frac{w\gamma}{2\pi})\frac{1}{2\pi}
\int^\infty_0 \rd\omega \exp[-4\omega^2t/\tau_0]\frac{\phi^2}{\omega^2 + \phi^4/16}.
\end{equation}
The Fourier transform of this expression, after some manipulation, can be written as
\begin{equation}
P(\tilde{Q}) = \pi\bar{t}^{1/2} \int^\infty_{-\infty} \rd y 
\frac{\re^{-\frac{\tilde{Q}^2}{4(A+|y|)} - y^2/\bar{t}}}{\sqrt{A+|y|}},
\end{equation}
where $\bar{t} = t/\tau_0$ and $A = \frac{\gamma^2}{\pi^2}\ln (a^{-1}/(\alpha M))$.
At small times, the integral sees its main contribution at small $y$, thus allowing
us to write
\begin{equation}
P(\tilde{Q})_{\bar{t} \ll 1} \approx \frac{\pi^{3/2}}{\sqrt{2A}}\exp\bigg[-\frac{\tilde{Q}^2}{4A}\left(1-\frac{\tilde{Q}^2\bar{t}}{32A^3}\right)\bigg].
\end{equation}
In this case the distribution is that of a static Gaussian in $\tilde{Q}$ with
small time dependent corrections.
At large times we can evaluate this integral in the saddle point approximation which yields 
\begin{equation}
P(Q(t))_{\bar{t}\gg 1} \sim \bigg(\frac{1}{\bar{t}\tilde{Q}^2}\bigg)^{1/6}
\exp\bigg[-\frac{3}{2^{5/3}}\bigg(\frac{\tilde{Q}}{\bar{t}^{1/4}}\bigg)^{4/3}\bigg].
\end{equation}
The reader may notice that quantum fluctuations 
play a significant role in modifying the Gaussian distribution of
the counting statistics seen in the simpler Ising-like reflectionless case.

\subsection{Drude Weight at arbitrary $\gamma$}

We can use the above expressions for the correlators to argue that there
is a zero frequency delta function piece to the conductivity. 
Expanding Eq.(\ref{S1}) in $\phi$ at small $\phi$ and differentiating 
it twice with respect to time, we obtain the correlation function for the 
current operators, $J_{\mu} = (\gamma/2\pi)\epsilon_{\mu\nu}\p_{\nu}\Phi$,
\begin{equation}
\la J(x,t)J(0,0)\ra = 
\frac{2}{\pi^2} \int dp v^2(p)\delta\left(x - t v(p)\right)\re^{-\beta\epsilon(p)} 
+ \mbox{non-ballistic piece} + \cdots,
\end{equation}
where $v(p) = \p\epsilon/\p p$. 
This current-current correlation function is valid for all of $\gamma$ as Eq. (\ref{S1})
is valid for all of $\gamma$ -- for $\gamma$ at a reflectionless point, this expression
reduces to one indicative of the Ising universality class.
From this we see that transport is universally ballistic in nature.  Using the fluctuation-dissipation 
theorem, we find the following finite Drude weight
\begin{equation}
\s(\omega) = 2\pi D(T)\delta(\omega) + ... , ~~ D(T) = \frac{\beta}{\pi^2}
\int \rd p v^2(p)\re^{-\beta\epsilon(p)}.
\end{equation}
We point out that the finding of a finite Drude weight in the sine-Gordon model is
consistent with the analysis of Ref. \cite{fuji1}
deducing a finite $D(T)$ in the
half-filled Hubbard model.  In this reference the 
thermodynamic Bethe ansatz equations with twisted
boundary conditions were studied. 
However a direct computation of the semi-classical current-current correlation functions yields a different result.  Both numerical and analytical calculations
(the latter based on the techniques
pioneered by Jepsen \cite{jepsen}) done in
Ref. \cite{garst} concluded that a semi-classical reduction of
an undoped sine-Gordon model at its reflectionless points has zero Drude weight.


\section{Discussion}

In this section we turn to a discussion of the discrepancies between the semi-classical results of
Refs. \cite{subir,kedarsubir,zarand}
and the form-factor computations presented in this work.  We believe that these discrepancies are
both real and represent differences
between the fully quantum theory and its semi-classical reduction.
In a quantum theory, conserved charges may be present which have no clear
analog in the semi-classical counterpart.
This is particularly noteworthy if the quantum theory possesses non-local conserved charges.  Two examples of such
theories are the O(3) non-linear sigma model (NLSM) and the sine-Gordon model.  The presence of these charges affects the
physics at {\it all} energy scales.  Thus we do not necessarily expect to find agreement (although agreement is
not precluded) between a semi-classical 
computation and the form-factor computation even in the case $T \ll M$.

Disagreement between the semi-classical treatment and the form-factor method is already known to exist
in the case of the O(3) NLSM.  This disagreement comes about in the low temperature computation of the 
zero magnetic field Drude weight, $D$,
for the spin conductivity.  The form-factor computation \cite{konik1}
tells us that the Drude weight is finite at low but finite temperature.  In contrast, the semi-classical
computation argues that the Drude weight is precisely zero.  The difference between the computations comes down
to the differences in conserved quantities between the quantum O(3) NLSM and its reductions to a classical
gas of particles interacting with an S-matrix, $-{\bf P}$.

The importance of conserved quantities to the finiteness of the Drude weight is encoded in Mazur's inequality.
As discussed in Ref. \cite{mazur}, $D$ is bounded from below by the inequality
\begin{equation}
D \ge c\sum_n \frac{\la JQ_n \ra^2}{\la Q_n^2 \ra},
\end{equation}
where $J$ is the relevant current operator, $Q_n$ are a set of orthogonal conserved quantities, and
$c$ is some constant.  All that is required then for a finite Drude weight is for a single matrix
element $\la J Q_n \ra$ not to vanish.  One can determine whether such matrix elements do necessarily vanish 
on the basis of the symmetries of the theory.  For example, under some discrete symmetry, we may
have 
$$
J \longrightarrow -J.
$$
Thus for the matrix element not to vanish one needs $Q_n$ simultaneously
to transform to $-Q_n$.  Using such arguments,
Ref. \cite{kedar} was able to argue that there existed no conserved quantity in the semi-classical reduction
whose corresponding matrix element $\la J Q_n \ra$ did not vanish.
However the situation is different in the quantum model.  In the O(3) NLSM, the spin current transforms as
a pseudo-vector.  Thus for a finite matrix element we need that there exists a 
corresponding charge to transform as a pseudo-vector.
Such a charge indeed does exist.  As discovered by L\"uscher \cite{luscher} there exists a series of non-local conserved
quantities transforming as such.  This series of conserved quantities has no 
obvious analog in the semi-classical theory.
On the basis of Mazur's inequality we are thus not surprised to find differences in the computation of the Drude
weight via form-factors and via semi-classics.

The form factor computation finding a finite Drude weight is supported by the computations of Ref. \cite{fuji2}.
In Ref. \cite{fuji2}, 
a non-zero Drude weight at finite temperature 
is extracted from a finite size scaling analysis of the thermodynamic Bethe ansatz equations for the O(3) NLSM
with twisted boundary conditions.  This calculation does not depend upon the form-factor architecture and so
is completely independent of the computation made in Ref. \cite{konik1}.

We find a similar story in the analysis of the sine-Gordon model.  As we demonstrated above, the sine-Gordon
model has a finite Drude weight for arbitrary $\gamma$ at small non-zero temperatures.  Furthermore this
is consistent with the analysis of the
thermodynamic Bethe ansatz equations of the half-filled Hubbard model \cite{fuji1} where a finite
Drude weight is found at finite temperature.  In contrast the semi-classical approach predicts
zero Drude weight at half-filling \cite{garst}.

Again the discrepancy between the quantum treatment of the sine-Gordon model and its semi-classical reduction
can be explained in terms of conserved quantities.
As is well understood \cite{leclair}, the sine-Gordon model has non-local currents 
that generate the quantum deformed SU(2) algebra
governing the theory.  Explicitly, these currents take the form
\begin{equation}
J^{non-local,R/L}_\pm = \re^{\pm \ri\sqrt{2}\phi_{L/R}/\gamma},
\end{equation}
where $\phi_{L/R}$ are the left/right components of the full Bose field, $\Phi$.
These currents couple to the current operator in a nontrivial fashion so that
$\la Q^{non-local}_{R/L}J\ra$ does not (obviously) vanish.  By Mazur's inequality, we
conclude that the Drude weight is finite.
However these non-local currents do not have obvious counterparts
in the reduction of 
the sine-Gordon model to a classical gas of two different particle types interacting
with an S-matrix, $-{\bf P}$.

Putting aside the consistency of the form-factor computations with the semi-classical picture, we can point
to an instance where the semi-classics appears to be internally inconsistent.
Let us focus on the reflectionless case of sine-Gordon.  
On one hand, at the reflectionless
points, the semiclassical expressions \cite{subir} 
for  $\la \exp(\ri\eta \Phi)\exp(-\ri\eta\Phi) \ra$, coincide with those derived by the formfactor approach.
Therefore by the same expansion in $\eta$ and differentiation in time that we employed in Section Vc, 
one can  deduce a finite Drude weight.
However as we have indicated, the 
direct computation of the semi-classical current-current correlation functions by Ref. \cite{garst} 
yields a different result.  Both numerical and analytical calculations
done in
this work concluded that a semi-classical reduction of
an undoped sine-Gordon model at its reflectionless points has zero Drude weight.
It is not clear how these two semi-classical computations can be resolved to one another.

A standard test of the appropriateness of the semi-classical approach is found in comparing
the thermal de Broglie wavelength $h/\sqrt{2\pi mT}$ with the average particle spacing
\cite{garst,subir}.  If the former is much smaller than the latter, the semi-classical approximation
is said to be good.  If the discrepancy between the form-factor computations and the analysis
of the thermodynamic Bethe ansatz equations is taken to be real, this test is inadequate.
Perhaps a better test may be found in the ability to integrate out the higher Matsubara modes of the
action of the model that appears in the appropriate path integral.  Using the semi-classical approximation
is equivalent to saying that quantum dynamics are unimportant to the physics of the problem.  Thus
in considering the model in imaginary time, one should be able to integrate out the non-zero Matsubara modes
leaving one with a dimensionally reduced but identical theory with possibly renormalized parameters.
This is what is done in Ref. \cite{halperin} where the $2+1$ dimensional O(3) NLSM is studied.
When the coupling $g$ is less than its critical value, one can integrate out the higher Matsubara modes
(at the one-loop level) and obtain a classical $2$ dimensional O(3) NLSM with renormalized
parameters.  However if this is tried on the $1+1$ O(3) NLSM, one finds that the quantum fluctuations
are considerably stronger and the one-loop computation is plagued by $T\rightarrow 0$ divergences.

In certain cases this may explain why the form-factor approach and the semi-classics agree as
say in the Ising model.  Agreement here is found because one can without difficulty integrate out the
higher Matsubara modes of a non-interacting model.
In the case of the sine-Gordon this explanation is less telling inasmuch as it is 
unclear how exactly to integrate
out the higher Matsubara modes in an expedient 
fashion (how exactly does one treat the cosine potential in a loop expansion, for example).
But in this case we can point to another difference
between the exact and the semi-classical analyses, namely
how the semi-locality index behaves in the presence of multiple particles.
As follows from the analysis conducted in Section III, 
with a non-diagonal S-matrix and a non-trivial semi-locality factor, 
the analysis of the residues of the poles in the form factors
depends on taking into account the presence of all particles.  
That is to say, the residue of a 2n-particle form factor does
not factorize simply into an n-fold product of the residues of 2-particle form factors.  
The analog to this behavior is not found
in the semi-classical analysis.


\section{Conclusions}
 
In this paper 
we have found universal expressions for the asymptotics of correlation 
functions of order parameter operators. 
These operators have $T=0$ nonzero vacuum expectation values 
thus breaking some discrete symmetry present in the theory. 
We restricted our consideration to theories with well defined asymptotic particle states 
(this excludes such models as the super sine-Gordon or the O(2N+1) Gross-Neveu model). 
Among such models, we have found two characteristic types of order parameter dynamics. 
One type appears in the models with either diagonal S-matrices (these include, among other models, 
the quantum Ising model and the sine-Gordon model 
at the reflectionless points) or for operators of non-diagonal theories with trivial semi-locality factors. 
These dynamics essentially coincide with the semi-classical 
dynamics described in \cite{subir}. 
Another type of dynamics is described by Eq. (\ref{S1}) and appears in the models 
where the zero momentum S-matrix corresponds to pure reflection and the order parameter has a non-trivial
semi-locality factor. 
These dynamics, as determined by the form-factors, differ dramatically from 
semi-classical treatments found in 
Refs. \cite{kedarsubir} and \cite{zarand}.  We believe this difference arises ultimately because
the conserved quantities of the fully quantum model and its semi-classical reduction differ
significantly.




\section{Acknowledgments}

This research was supported by DARPA under the QuIST Program, by the 
Institute for Strongly Correlated and Complex Systems at BNL (BLA), and by 
the DOE under contract number DE-AC02 -98 CH 10886 (RMK, AMT).  
All the authors are grateful to the International Centre for Theoretical
Physics in Trieste, Italy, for its hospitality during which this project was
completed.
AMT is grateful to
Princeton University for hospitality and to S. A. Lukyanov for 
attracting his attention to Ref. \cite{kiev}. 
We are also grateful to H. Babujian, M. J. Bhaseen, K. Damle, F. H. L. Essler, V. A. Fateev,  
V. E. Korepin, S. Sachdev, F. A. Smirnov, and G. Zarand for fruitful discussions and interest in the work. 



 

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