The 1D Bose Gas with Weakly Repulsive Delta Interaction
\textsc{M.T. Batchelor}$^{1}$
\textsc{X.W. Guan }$^{1}$
cond-mat0412645.tex
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\title{The 1D Bose Gas with Weakly Repulsive Delta Interaction}

\author{\textsc{M.T. Batchelor}$^{1}$,
 \textsc{X.W. Guan }$^{1}$, \textsc{C. Dunning}$^{2}$ and  \textsc{J. Links}$^{2}$
%\thanks{E-mail address: abc@def.com} and Author \textsc{Name}$^{3}$\thanks{Present address: Department of Applied Physics, University of Tokyo, Tokyo.}
}

\inst{$^{1}$    Department of Theoretical Physics, RSPSE, and\\
Mathematical Sciences Institute\\
The Australian National University, Canberra ACT 0200, Australia\\
$^{2}$ Centre for Mathematical Physics, School of Physical Sciences,\\
The University of Queensland, Brisbane 4072, Australia}

\abst{
We consider the asymptotic solutions to the Bethe ansatz equations of the
integrable model of interacting bosons in the weakly interacting limit.
%
In this limit we 
establish that the ground state maps to the highest energy 
state of a strongly-coupled repulsive bosonic pairing model.  
}

\kword{Integrable models, 1D Bose gas, Bethe ansatz equations, BCS pairing models}


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%\def\a{\alpha}
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\def\e{\epsilon}
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%\def\l{\lambda}
\def\l{\left}
\def\r{\right} 
\def\o{\omega}
\def\t{\theta}
\def\s{\sigma}
%\def\D{\Delta}
\def\ap{\approx} 
\def\ve{\varepsilon}

% Shorthands for \begin{equation} and the like

\def\beq{\begin{equation}}
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\def\ba{\begin{eqnarray}}
\def\ea{\end{eqnarray}}
\def\no{\nonumber}
%\def\le{\langle}
\def\re{\rangle}
\def\lt{\left}
\def\rt{\right}
\def\o{\omega}
\def\d{\dagger}
\def\nn{\nonumber} 
\def\j{{ {\cal J}}}
\def\n{{\hat n}}
\def\N{{\hat N}}
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\def\B{\mathcal{B}}
\def\C{\mathcal{C}}
\def\D{\mathcal{D}}
\def\R{\mathcal{R}}
\def\Y{\mathcal{Y}}
\def\Z{{\mathbb{Z}}}
\def\I{\mathcal{I}}
\def\T{\mathcal{T}}
\def\E{\mathcal{E}}
\def\G{\mathcal{G}}
\def\GG{\overline{\mathcal{G}}}
\def\FF{\overline{\mathcal{F}}}
\def\FFF{\overline{\overline{\mathcal{F}}}}
\def\P{\mathcal{P}}
\def\PP{\overline{\mathcal{P}}}
\def\S{\mathcal{S}}
\def\L{\mathcal{L}}
\def\W{\mathcal{W}}
\def\F{\mathcal{F}}
\def\KK{\mathcal{K}}
\def\KKK{\overline{\mathcal{K}}}
\def\Q{\mathcal{Q}}
\def\QQ{\overline{\mathcal{Q}}}
\def\aa{{\mathpzc{a}}}
\def\dd{{\mathpzc{d}}}
\def\g{{\mathfrak{g}}}
\def\cC{{\mathfrak{C}}}
\def\Y{\mathcal{Y}}
\def\j{\mathcal{J}} 
\def\n{{\hat n}}
\def\N{{\hat N}}
\def\La{\Lambda}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\maketitle

\section{Introduction} %% No sections necessary for express letters, letters and short notes

There has been a revival of interest in the exactly solved 1D model of
interacting bosons \cite{LL}.
%
In part this is due to the experimental realisation of a
quasi-1D quantum gas of bosons at ultracold 
temperatures.\cite{EXP1,EXP2,EXP3,EXP4,EXP5}
%
The essential point is that the interactions between the trapped atoms
can be tuned to bring about a continuous passage from the weakly interacting regime
to the strongly interacting regime.
%
In this way the full subtleties of quantum many-body physics are observed, from 
Bose-Einstein condensation in the weak coupling regime to the
pronounced fermionic behaviour of the Tonks-Girardeau gas in  
the strong coupling regime.

Recently it was observed \cite{bgm} with the help of numerical
analysis that the Bethe ansatz roots for the ground state of the
exactly solved 1D Bose gas in the weak coupling limit satisfy 
a similar set of equations as an exactly solved BCS boson pairing model
in the strong coupling limit.
%
Here we further clarify this correspondence in two ways:
first we derive, for arbitrary particle number, the system of
equations satisfied by the ground state roots of the 1D Bose gas in the
weak coupling limit.  
%
Then we show that the mapping to a BCS type
system is precise for certain bosonic systems where the Cooper pairs
are formed from integer spin particles.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weak coupling limit}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Hamiltonian 
%
\bea &&{\cal H} =-\sum_{j=1}^N \frac{\partial^2}{\partial x_j^2}
+2c\sum_{j<k}^N \delta(x_j-x_k)   
\eea
%
describing $N$ interacting bosons on a periodic interval of length $L$ has been
extensively studied  \cite{LL,L, MC, YY, Gaudin, MAT, KOR}.
%
The eigenstates have energy and momenta  given by
% 
\beq
{\cal E}=\sum_{j=1}^N k_j^2, \qquad 
P=\sum_{j=1}^N k_j,
\eeq
where $k_j$ satisfy the Bethe ansatz equations (BAE) 
\beq
\exp( \mathrm{i} k_jL)=-\prod_{l=1}^N\frac{k_j-k_l+\mathrm{i}c}
{k_j-k_l-\mathrm{i}c}, \qquad j=1,\ldots,N. \label{BA}  
\eeq   
For repulsive interactions ($c>0$) it is known that the Bethe roots $k_j$ 
are real  and distinct \cite{KOR}. 
%
Moreover, the eigenspectrum is positive, i.e., 
${{\cal E}_0} \le {\cal E}_1 \le {\cal E}_2 \dots$, where ${\cal E}_0$ is the
ground state.


Asymptotic solutions to the BAE (\ref{BA}) are possible in two limiting cases: 
$Lc\ll1$ and $Lc\gg1$, corresponding to weak and strong delta interaction.  
%
In the limit $Lc\gg1$ \cite{bgm}, which we do not discuss here, 
the asymptotic solutions to the BAE (\ref{BA}) describe the Tonks-Giradeau gas \cite{Tonks,KG}.
%
In the limit $Lc\ll1$, numerical checks suggest that the momenta $k_j$ are proportional to the square root of $c$ \cite{bgm}.
%
It follows that,  to order $c^2$, the BAE (\ref{BA}) reduce to 
%
\begin{eqnarray}
\exp( \mathrm{i} k_jL)&\approx&1-2\sum^{N}_{\ell=1}\frac{c^2}{(k_j-k_{\ell})^2}-4\sum_{\ell=1}^{N-1}\sum_{\ell < \ell '=2}^{N}\frac{c}{(k_j-k_{\ell})}\frac{c}{(k_j-k_{\ell '})}
\nonumber\\
& &
+ \,\, \mathrm{i} \sum_{\ell=1}^N\frac{2c}{(k_j-k_{\ell})},
\end{eqnarray}
%
in which the summations exclude $\ell=j$ and $\ell'=j$. 
%
Indeed, to order $c^2$,
%
\begin{eqnarray}
\cos(k_j L)&\approx & 1-2\sum^{N}_{\ell=1}\frac{c^2}{(k_j-k_{\ell})^2}-4\sum_{\ell=1}^N\sum_{\ell < \ell '=2}^{N}\frac{c}{(k_j-k_{\ell})}\frac{c}
{(k_j-k_{\ell '})},\label{BE-app1}\\
\sin(k_j L)& \approx & \sum_{\ell=1}^N\frac{2c}{(k_j-k_{\ell})}.
 \label{BE-app2}
\end{eqnarray}
%
The solution of which determines the asymptotic roots of the BAE (\ref{BA}).
%
>From Eqs. (\ref{BE-app1}) and (\ref{BE-app2}) the Bethe roots are seen to satisfy 
%
\begin{equation}
k_j=\frac{2\pi d_j}{L}+\frac{2c}{L} \, {\sum_{\ell\ne j}^N} \frac{1}{k_j-k_\ell}, 
\qquad j=1,\ldots N.
\label{momenta}
\end{equation}
%
Here $d_j=0,\pm 1,\pm 2, \ldots$
denotes excited states and the summation excludes $j=\ell$. 

The asymptotic equations are closely related to those appearing in 
Stieltjes problems \cite{SD}. 
%
The ground state has zero total momentum with $d_j=0$ for $j=1,\ldots,N$, with
the ground state energy per particle 
%
\begin{equation}
\frac{{\cal E}_0}{N}=\frac{c(N-1)}{L},
\label{enrgy1}
\end{equation}
%
following directly from Eq.  (\ref{momenta}).
%
The algebraic equations (\ref{momenta}) for the ground state
are given in Gaudin \cite{Gaudin}.
%
In this way Gaudin showed that the $k_j$ are roots
of Hermite polynomials of degree $N$, namely $H_N(k)=0$.
%
They are also related to roots of the Laguerre polynomial.\cite{bgm}
%
These connections provide a systematic way for studying
quantities such as the momentum distribution function and correlations.
%
The normalised momentum density distribution is given by the semi-circle law 
\cite{Gaudin}
\begin{equation}
n(k)=\frac{L}{2N\pi c}\left(4c\rho-k^2\right)^{\frac{1}{2}}, 
\label{density}
\end{equation}
where $\rho=N/L$.
%
The stronger the interaction strength, the larger the momentum distribution region. 
%
This reveals a significant signature of the 1D Bose gas in the weakly repulsive regime $Lc\ll1$. 
%
Remarkably, this behaviour was recently observed in the
experiments for weakly interacting bosons.\cite{EXP1,EXP2,EXP3,EXP4,EXP5}
%
If $c=0$ all the particles condense in the ground state at zero temperature.


Now consider the excitations above the ground state, which have 
total momentum $P={2n\pi}/{L}$ with $n=0,\pm 1,\pm 2, \ldots$.
%
As an example of the solutions obtained from the asymptotic equations 
(\ref{momenta}), consider the numerical data for four bosons given in  {Table I}.
%
Each state is characterised by the quantum numbers $d_j$, with
total momentum $P=\sum_j^N {2d_j\pi}/{L}$.
%
The excitation energies are at least doubly degenerate in view of the
counterpart assignments $- d_j$ with momenta $- k_j$.
%
Shown for direct comparison are the numerical solutions from the full BAE (\ref{BA}).
%
Clearly the agreement is excellent.

Let us consider the lowest excited state ${\cal E}_1$, with total 
momentum $P={2\pi }/{L}$, in more detail. 
%
Without loss of generality,  we choose an assignment $d_1=1$ and $ d_j=0$ for $j=2,\ldots, N$. 
%
Using Eq.~(\ref{momenta}) we can approximately calculate the lowest excitation energy.
%
Specifically, 
\begin{equation}
\frac{{\cal E}_1}{N} =\sum_{j=1}^Nk_j^2
\,\, \approx  \,\, \frac{2\pi k_1}{LN} + \frac{c(N-1)}{L}, 
\end{equation} 
from which we can infer that  
\begin{eqnarray}
k_1& \approx & \frac{2\pi}{L}+\frac{c(N-1)}{\pi},\label{k1} \\
\frac{{\cal E}_1}{N} &\approx & \frac{c(N-1)}{L}+\frac{4\pi^2}{L^2N}+\frac{2c(N-1)}{LN}, 
\label{E1}
\end{eqnarray}
%
which are valid for $N/L$ finite.
%
These approximations are superior to those given earlier \cite{bgm}.
%
It is clearly seen from Eq.~(\ref{E1}) that the energy gap will vanish in the limit $N \to \infty$. 
%
For the values of Table I, 
Eqs.~(\ref{k1}) and (\ref{E1}) give $k_1\approx 6.30705$ and 
${{\cal E}_1} \approx 39.9284$, which are
in good agreement with the numerical results obtained from 
Eqs.~(\ref{momenta}) and (\ref{BA}). 
%
As a further example, consider $N=8$ with $c=0.025$ and $L=2$.
%
In this case Eqs.~(\ref{k1}) and (\ref{E1}) give  $k_1\approx 3.1973$
and ${{\cal E}_1} \approx 10.7446$.
%
These results are to be compared with $k_1\approx 3.1693$ and 
${{\cal E}_1} \approx 10.3065$,
which follow from Eq.~(\ref{momenta}), and  $k_1\approx 3.1966$ and
${{\cal E}_1} \approx 10.7403$, obtained from the BAE (\ref{BA}). 
 

\begin{center}
\begin{table}
\caption{The ground state and the leading five excitations of four weakly
interacting bosons with $L=1$ and $c=0.025$. 
%
For each state, solutions of the asymptotic equations (\ref{momenta}) 
are shown on the first line.
%
The second line shows solutions obtained from the full BAE (\ref{BA}).
}
\vskip 3mm
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 
$n$&${\cal E}_n$&$P$&$k_4$&$k_3$&$k_2$&$k_1$&$(d_1,d_2,d_3,d_4)$\\
\hline 
0&
0.30000&
0&
-0.36910&
-0.11731&
 0.11731& 
0.36910 &
(0,0,0,0)
\\
0&
0.29938&
0&
-0.36872&
-0.11719&
0.11719& 
0.36872 &
\\
\hline
1&
39.88783&
2$\pi $&
-0.28162&
-0.00793& 
0.26576&
6.30379&
(1,0,0,0)
\\
1&
39.92754&
2$\pi $&
-0.28133&
-0.00792&
 0.26548& 
6.30697&
\\
\hline
2&
79.37604&
4$\pi $&
-0.17378&
0.14205&
6.13794&
 6.45378& 
(1,1,0,0)
\\
2&
79.45587&
4$\pi $&
-0.17361&
0.14190&
6.14129&
6.45679& 
\\
\hline
3&
79.42614&
0 &
-6.29985&
-0.15791&
0.15791&
 6.29985& 
(1,-1,0,0)
\\
3&
79.50607&
0 &
-6.30303&
-0.15775&
0.15775&
6.30303& 
\\
\hline
4&
118.7646&
6$\pi $&
-0.02380&
6.01424&
6.28793&
 6.56162& 
(1,1,1,0)
\\
4&
118.8844&
6$\pi $&
-0.02378&
6.01771&
6.29111&
6.56452& 
\\
\hline
5&
118.8647&
2$\pi$ &
-6.29590&
-0.00793&
6.13393&
6.44990& 
(1,1,-1,0)
\\
5&
118.9846&
2$\pi$ &
-6.29907&
-0.00793&
6.13727&
6.45292& 
\\
\hline 
\end{tabular}
\end{table}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\twobeone
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Link to the BCS model}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The weak coupling limit provides a direct link between the integrable 1D Bose gas and 
the integrable BCS pairing models.
%
In this regime the Bethe roots
for the ground state are of the form
%
%\\ {\bf \large OK ?} 
\begin{equation}
k_1 = -k_2 =\sqrt{E_1},\,k_3=-k_4=\sqrt{E_2},\ldots, \, 
k_{2M-1}=-k_{2M}=\sqrt{E_M},\label{P3}
\end{equation}
%
where the $E_i$ satisfy the equations (cf (\ref{momenta}))
%
\begin{equation}
-\frac{L}{2c} + {\sum_{j\ne i}^{M}} \frac{2}{E_i-E_j}=-\frac{1}{2E_i},
\label{bose}
\end{equation}
%
for $ i = 1, \ldots, M$ and  the total number of bosons are even, i.e. $N=2M$. 

For an odd number of  bosons, $N=2M+1$, the ground state Bethe roots are given by
%
%\\ {\bf \large OK ?}
\begin{equation}
k_1=0,\,k_{2}=-k_{3}=\sqrt{E_1},\,k_{4}=-k_{5}=\sqrt{E_2},
\ldots, k_{2M}=-k_{2M+1}=\sqrt{E_M},\label{P4}
\end{equation}
%
where the ${E_i}$ satisfy the equations
%
\begin{equation}
{\sum_{j\ne i}^{M}} \frac{2}{E_i-E_j}-\frac{L}{2c}=-\frac{3}{2E_i},
\label{bose2}
\end{equation}
%
for $ i = 1, \ldots, M$.

Similar equations have arisen in a number of contexts \cite{BCS1,yba,BCS3,BCS5,SD}.
%
Of particular interest here is the connection between Eq. (\ref{bose})
and Richardson's equations for the BCS pairing model \cite{BCS3,BCS4}
in the strong coupling limit \cite{yba}.  
%
We now make the connection with these `pairing interaction' Hamiltonians more precise. 
%
In terms of the $su(2)$ algebra with commutation relations 
\beq [S^z, S^{\pm}] = \pm S^\pm, \qquad  [S^+,S^-] =2 S^z, \label{su2} 
\eeq
we consider the class
of Hamiltonians acting on the $\L$-fold tensor product of
$su(2)$-modules, (not necessarily finite-dimensional,) with lowest
weight $-\S$.
%
Specifically,
\bea 
{\cal H}&=& \sum_{j=1}^{\L}2\e_j(S_j^z+\S
I)-g\sum_{j,k=1}^{\L}S_j^+S_k^-,
\label{ham} 
\eea
where $I$ is the identity operator.
%
The single particle energy levels $\e_j$ and the coupling constant $g$ are arbitrary real  
parameters.
%
The energy levels of the
Hamiltonians (\ref{ham}) are ${\cal E}=2\sum_{j=1}^m E_j$ where the
$\{E_j\}$ satisfy the Bethe equations \cite{lzmg}
\beq
\frac{2}{g}+ \sum_{k=1}^{\L}\frac{2\S}{E_j-\e_k}
=\sum_{\ell\neq j}^m \frac{2}{E_j-E_\ell}.
\label{bcsbae} 
\eeq

Suppose each energy level $\e_j$ can be occupied by a
spin $s$ particle so the degeneracy is $2s+1$. We introduce the creation and
annihilation operators $a_{\sigma},\,a_{\sigma}^\dagger$, where
$\sigma=-s,-s+1,...,s$ is the spin label, satisfying the usual
boson commutation (resp. fermion anticommutation) relations for integer (resp.
half-odd integer) values of $s$. 
We can define a representation
of $su(2)$ through
\bea && 
S^-=-\frac{1}{2}\sum_{\sigma=-s}^s 
(-1)^{s-\sigma}a_{\sigma}a_{\overline{\sigma}},
~~~
S^+=\frac{1}{2}\sum_{\sigma=-s}^s (-1)^{s-\sigma} a_{{\sigma}}^{\dagger}
a_{\overline{\sigma}}^{\dagger},
\nn \\  
&&
S^z=\frac{1}{4}\sum^s_{\sigma=-s}\left(2a^\dagger_{\s}a_{\s}
+(-1)^{2s} I\right), \label{idr} 
\eea   
satisfying (\ref{su2}) where $\overline{\sigma}=-\sigma$. 
%
In each case the vacuum state $\left|0\right>$ is a lowest weight state
of weight $-\S=(-1)^{2s} \frac14 (2s+1)$. 
%
In the fermionic case (where $s$ is a half-odd integer) these
representations are finite-dimensional  and $\S=\frac12,1,\frac32,\dots$. 
%
In the bosonic case (where $s$ is an integer) the representations are
infinite-dimensional and 
$ \S=-\frac14,-\frac34,-\frac54,\dots$. For the bosonic case, we can also consider
lowest weight states of the form 
\beq 
a^\dagger_{\sigma_1}\left|0\right>, \qquad
a^\dagger_{\sigma_1} a^\dagger_{\sigma_2} \left|0\right> , \qquad
a^\dagger_{\sigma_1} a^\dagger_{\sigma_2} a^\dagger_{\sigma_3}
\left|0\right>, \qquad {\rm etc},
\label{bosons}  
\eeq 
provided $\sigma_i\neq \overline{\sigma_j}$ for all $i,\,j$. 
%
When there are
$z$ bosons in a lowest weight state of the form (\ref{bosons}) 
the lowest weight is given by 
$-\S=\frac14 (2z+(-1)^{2s}(2s+1))$ 
giving rise to the sequence 
$\S=-\frac12 z-\frac14,-\frac12 z-\frac34,-\frac12 z-\frac54,\dots$.  
Under these representations (\ref{ham}) describes  pairing Hamiltonians for
spin $s$ particles, and the integer $m$ in (\ref{bcsbae}) denotes the number
of Cooper pairs in the corresponding eigenstate.
% 
The usual reduced BCS model for spin $\frac12$ fermions corresponds to the case
$s=\frac12$ giving $\S=\frac12$ \cite{lzmg,yba}. 


Following \cite{yba} we consider the large $g$ limit. Let $r\leq m$ denote
the number of roots $E_j$ which diverge as $g\rightarrow \infty$.  
For these roots we have to lowest order in $g^{-1}$  
\beq
\frac{2}{g}+ \frac{n}{E_j}  
=\sum_{\ell\neq j}^r \frac{2}{E_j-E_\ell}.
\label{limbae}  \eeq
where $j=1,\ldots,r$ and
\beq n=2\S\L+2r-2m.  \label{rel} 
\eeq  
%{\bf \large OK ?}\\ 
The remaining roots $E_j$, $j=r+1,\ldots,m$, are small in comparison with $g$.
%
Comparison between      
(\ref{bose}), (\ref{bose2}) and (\ref{limbae}) shows we may identify 
%\\ {\bf \large OK ?}
$E_j=k_{2j-1}^2=k_{2j}^2$ for $N$ even and 
$E_j=k_{2j}^2=k_{2j+1}^2$ for $N$ odd, with 
\beq
\quad \frac{2}{g}=\frac{L}{2c}, \quad r=M,
\eeq
where $n=-\frac{1}{2}$ for $N$ even, and $n=-\frac{3}{2}$ for $N$ odd. 
%
The fact that $\L,\,m$ are positive integers, $r$ is a non-negative
integer and $r\leq m$ imposes severe restrictions on the allowed
solutions of (\ref{rel}). 
%
For $n=-\frac12$ the only solution is $\L=1,\,m=M$ and $\S=-\frac14$ ($s=0,\,z=0$). 
%
For $n=-\frac32$ we can have $\L=3,\,m=M$ and $\S=-\frac14$ ($s=0,\,z=0$).  
%
In all cases $s$ is an
integer so the mapping is to a bosonic pairing model. Moreover, since
$g$ is positive, and the bosonic representations (\ref{idr}) are
non-unitary (specifically $(S^+)^\dagger =-S^-$) these systems
describe {\it repulsive} pairing interactions. 
%
In the strong coupling limit, all energy levels may collapse into one
multiply-degenerate level for $N$ even and into three energy levels for $N$ odd.
%
In comparison with large pairing scattering energies, the level spacing is negligible. 
%
Therefore the multiply-degenerate levels are a reasonable expectation 
in the strong coupling limit.

Multiplying each of (\ref{limbae}) by $E_j$ and taking the sum gives 
\bea
E(r)=\sum_{j=1}^r E_j&=&gr(r-n-1) \nn \\ &=&-gr(2\S\L+r-2m+1).
\eea 
%{\bf \large OK ?}\\ 
The energy  
function $E(r)$ has zeroes at $0$ and $2m-2\S\L-1$ and attains the maximum
value at $m-\S\L-\frac12$. 
%\\ {\bf \large OK ?}\\
Note that for bosonic pairing models 
$-\S\geq \frac14 +\frac{z}{4}$ ($z=0$ in this case), 
which imposes the lower bound $m-\frac14$ for the value at which the
maximum occurs. The energy gaps between successive levels are found to be
\bea 
\Delta(r)&=&E(r)-E(r+1)  \nn \\
&=& 2g(\S\L+r+1-m), 
\label{gaps} 
\eea
and in particular 
\bea 
\Delta({m-1})&=&2g\S\L,    \nn \\
\Delta({m-2})&=&2g(\S\L-1),    \nn \\
\Delta({m-3})&=&2g(\S\L-2).     
\eea 
%
%{\bf \large OK ?}\\ 
We see that $\Delta(r) < 0$ if $r \in \{0,1,\ldots,m-1 \}$ and 
$\S\le -\frac{z}{2}-\frac14$.
%\\ {\bf \large OK ?}\\  
Thus $E(r)$ takes its maximal value when $r=m=M$.
%
%\\ {\bf \large OK ?}\\
%
>From Eqs.~(\ref{bose}) and (\ref{bose2}), the
ground state energy per particle of the weakly interacting Bose gas follows as 
%
\begin{equation}
\frac{{\cal E}_0}{N}=\sum_{i=1}^{N}\frac{k_i^2}{N}=\frac{2 E(M)}{N}=\frac{c(N-1)}{L}.
\end{equation}
%
This agrees with Eq.~(\ref{enrgy1}) and the results of \cite{LL,TAK,Wadati} among others.
%
We see that it also coincides with the highest energy per particle state
of the strongly coupled BCS boson pairing model (\ref{ham}).

In conclusion we have considered the asymptotic solutions to the Bethe ansatz equations for the 
the weakly interacting 1D Bose gas. 
%
We have established that the ground state 
%of the weakly interacting 1D boson gas
maps to the highest energy 
state of a strongly-coupled repulsive bosonic pairing model.  
%
However, the precise link between the integrable boson model with weakly repulsive 
delta interaction and the standard BCS fermionic model is quite subtle and  
deserves further investigation.


\section*{Acknowledgment}

This work has been supported by the Australian Research Council.
%
MTB thanks the organisers of The International Conference on the Statistical Physics of Quantum Systems for the opportunity to present this work. 
%
We thank Huan-Qiang Zhou  for helpful discussions
and the referee for some helpful remarks.


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\end{thebibliography}


\end{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Experimental}

\begin{figure}[tb]
%\begin{center}
%\includegraphics{}
%\end{center}
\caption{Basically, figures and tables must be located near the place where they appear for the first time in the text.}
\label{f1}
\end{figure}

\section{Discussion}

\subsection{Subsection sample 1}

\begin{table}[tb]
\caption{Table caption.}
\label{t1}
\begin{tabular}{ccc}
\hline
abc & def & ghi \\
\cline{2-3}
jkl & mno & pqr \\
\hline
\end{tabular}
\end{table}

\begin{table}[tb]
\caption{Table caption.}
\label{t2}
\begin{tabular}{ccccc}
\hline
abc & \multicolumn{4}{c}{defgh} \\
ijk & lmn & opq & rst & uvw \\
xyz \\
\hline
\end{tabular}
\end{table}

\subsubsection{Subsubsection sample 1}

\subsubsection{Subsubsection sample 2}

You can use \verb|\section| $\to$ \verb|\subsection| $\to$ \verb|\subsubsection| $\to$ \verb|\paragraph| $\to$ \verb|\subparagraph|.  Neither the \verb|\paragraph| nor the \verb|\subparagraph| are labelled with numbers.

\subsection{Subsection sample 2}

\section{Conclusions}

\section*{Acknowledgment}

\appendix
\section{Sample}

Equations in the appendix will be numbered as (A$\cdot$1), (A$\cdot$2), (A$\cdot$3) \ldots.

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\end{thebibliography}

\end{document}


cond-mat0412645.extracted_bib
\begin{thebibliography}{999}

\bibitem{LL} E.H. Lieb and W. Liniger:  Phys. Rev. \textbf{130} (1963) 1605.

\bibitem{EXP1} H. Moritz, T. S\"oferle, M. K\"ohl and T. Esslinger: 
Phys. Rev. Lett. \textbf{91} (2003) 250402.

\bibitem{EXP2} T. S\"oferle, H. Moritz, C. Schori, M. K\"ohl and T. Esslinger: 
Phys. Rev. Lett. \textbf{92} (2004) 130403.

\bibitem{EXP3} B.L. Tolra et al.: Phys. Rev. Lett. \textbf{92} (2004) 190401.

\bibitem{EXP4} B. Paredes et al.: Nature \textbf{429} (2004) 277.

\bibitem{EXP5} T. Kinoshita, T. Wenger and D.S. Weiss: Science \textbf{305} (2004) 1125.

\bibitem{bgm} M. T. Batchelor, X. W. Guan  and  J. B. McGuire: J. Phys. A  \textbf{37} (2004) L497.

\bibitem{L} E.H. Lieb: Phys. Rev. \textbf{130} (1963) 1616.

\bibitem{MC} J.B. McGuire: J. Math. Phys. \textbf{5} (1964) 622.

\bibitem{YY} C.N. Yang and C.P. Yang: J. Math. Phys. \textbf{10} (1969) 1115.

\bibitem{Gaudin}M. Gaudin: Phys. Rev. A \textbf{4} (1971) 386.

\bibitem{MAT} D.C. Mattis: 1993 {\em The Many-Body Problem} (World Scientific, Singapore).

\bibitem{KOR} V.E. Korepin, A.G. Izergin and N.M. Bogoliubov: 1993 
{\em Quantum Inverse Scattering Method and Correlation Functions}
(Cambridge University Press, Cambridge).

\bibitem{Tonks} L. Tonks: Phys. Rev. \textbf{50} (1936) 955.

\bibitem{KG} M. Girardeau: J. Math. Phys. \textbf{1} (1960) 516.

\bibitem{BCS1} R.W. Richardson: Phys. Lett. \textbf{3} (1963) 277;  Phys. Lett. 
\textbf{5} (1963) 82. \newline
R.W. Richardson and N. Sherman: Nucl. Phys. \textbf{52} (1964) 221;  Nucl. Phys.  \textbf{52} (1964) 253.

\bibitem{yba} E. A. Yuzbashyan, A. A.  Baytin and B. L. Altshuler: 
 Phys. Rev. B \textbf{68} (2003) 214509.
\bibitem{BCS5}J. Dukelsky, S. Pittel and G. Sierra: Rev. Mod. Phys. \textbf{76} (2004) 643.

\bibitem{SD} B.S. Shastry and A. Dhar: J. Phys. A \textbf{34} (2001) 6197.

\bibitem{BCS3} J. von Delft and D.C. Ralph: Phys. Rep. \textbf{345} (2001) 61.

\bibitem{BCS4} G. Sierra:   Nucl. Phys. B \textbf{572}  (2000) 517.\newline
L. Amico, A. Di Lorenzo and A. Osterloh:  Phys. Rev. Lett. \textbf{86} (2001) 5759.\newline
H.-Q. Zhou, J. Links, R.H. McKenzie and M.D. Gould: Phys. Rev. B \textbf{65}  (2002) R060502.

\bibitem{lzmg} J. Links, H.-Q. Zhou, R. H. McKenzie and M. D. Gould: 2003 
 J. Phys. A: Math. Gen.  \textbf{36} (2003) R63.

\bibitem{TAK} M. Takahashi: Prog. Theor. Phys. \textbf{53} (1975) 386. 

\bibitem{Wadati} M. Wadati: J. Phys. Soc. Jpn \textbf{71} (2002) 2657.

\end{thebibliography}