diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000000000000000000000000000000000000..3ccafb42187cb1b6b15af521ae9e44fe4f6593e2
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,14 @@
+*~
+.DS_Store
+
+*.aux
+*.log
+*.out
+*.toc
+
+*.synctex.gz
+
+by-eps*
+cc-eps*
+cc.*
+cc_large*
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diff --git a/LICENSE b/LICENSE
new file mode 100644
index 0000000000000000000000000000000000000000..10fabd90118f7ce38bb2e4753e105e744442a429
--- /dev/null
+++ b/LICENSE
@@ -0,0 +1,395 @@
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diff --git a/README.md b/README.md
index df5b2866fca2b850cd097462c47da6457064a8ff..7aeecb056e2eb03a8d183d11fa0e7ca17fb072b9 100644
--- a/README.md
+++ b/README.md
@@ -1,3 +1,27 @@
-# SciPostPhys.1.1.001
+# Source files for [SciPostPhys.1.1.001](https://scipost.org/SciPostPhys.1.1.001)
-Source files for [SciPostPhys.1.1.001](https://doi.org/10.21468/SciPostPhys.1.1.001).
\ No newline at end of file
+Source files for
+
+_Quantum quenches to the attractive one-dimensional Bose gas: exact results_
+
+Lorenzo Piroli, Pasquale Calabrese, Fabian H. L. Essler
+
+Published as doi:[10.21468/SciPostPhys.1.1.001](https://doi.org/10.21468/SciPostPhys.1.1.001).
+
+
+## License
+
+© L. Piroli et al.
+
+Published by the SciPost Foundation.
+
+Shield: [![CC BY 4.0][cc-by-shield]][cc-by]
+
+This work is licensed under a
+[Creative Commons Attribution 4.0 International License][cc-by].
+
+[![CC BY 4.0][cc-by-image]][cc-by]
+
+[cc-by]: http://creativecommons.org/licenses/by/4.0/
+[cc-by-image]: https://i.creativecommons.org/l/by/4.0/88x31.png
+[cc-by-shield]: https://img.shields.io/badge/License-CC%20BY%204.0-lightgrey.svg
diff --git a/SciPost.cls b/SciPost.cls
new file mode 100644
index 0000000000000000000000000000000000000000..b4b11325e0ca50dca89d99d3c272612d718fc545
--- /dev/null
+++ b/SciPost.cls
@@ -0,0 +1,87 @@
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesClass{SciPost} % SciPost Latex Template v1a (2016/06/14)
+
+
+\LoadClass[11pt,a4paper]{article}
+
+
+% Layout
+\RequirePackage[top=12mm,bottom=12mm,left=30mm,right=30mm,head=12mm,includeheadfoot]{geometry}
+\bigskipamount 6mm
+
+% For table of contents: remove trailing dots
+\RequirePackage{tocloft}
+\renewcommand{\cftdot}{}
+% Add References to TOC
+\RequirePackage[nottoc,notlot,notlof]{tocbibind}
+
+
+% Spacings between (sub)sections:
+\RequirePackage{titlesec}
+\titlespacing*{\section}{0pt}{1.8\baselineskip}{\baselineskip}
+
+
+% Unicode characters
+\RequirePackage[utf8]{inputenc}
+
+% doi links in references
+\RequirePackage{doi}
+
+% Math formulas and symbols
+%\RequirePackage{amsmath,amssymb} % Redundant (clashes) with mathdesign
+\RequirePackage{amsmath}
+
+% Hyperrefs
+\RequirePackage{hyperref}
+
+% Include line numbers in submissions
+\RequirePackage{lineno}
+
+% SciPost BiBTeX style
+\bibliographystyle{SciPost_bibstyle}
+
+% SciPost header and footer
+\RequirePackage{fancyhdr}
+\pagestyle{fancy}
+
+\makeatletter
+ \let\ps@plain\ps@fancy
+\makeatother
+
+\RequirePackage{xcolor}
+\definecolor{scipostdeepblue}{HTML}{002B49}
+
+\DeclareOption{submission}{
+
+\lhead{
+% \colorbox{scipostdeepblue}{\strut \bf \color{white} ~Submission }
+ \colorbox{scipostdeepblue}{\strut \bf \color{white} ~SciPost Physics }
+}
+
+\DeclareOption{LectureNotes}{
+ \lhead{
+ \colorbox{scipostdeepblue}{\strut \bf \color{white} ~SciPost Physics Lecture Notes }
+ }
+}
+\ProcessOptions\relax
+
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+ \colorbox{scipostdeepblue}{\strut \bf \color{white} ~Submission }
+}
+}
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+
+
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+
+\lhead{
+
+}
+
+\rhead{
+ \colorbox{scipostdeepblue}{\strut \bf \color{white} ~SciPost Physics }
+}
+
+}
+\ProcessOptions\relax
+
diff --git a/SciPostPhys_1_1_001.pdf b/SciPostPhys_1_1_001.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..962e0f34286a4748e457eec59bc494b97e0bb94d
Binary files /dev/null and b/SciPostPhys_1_1_001.pdf differ
diff --git a/SciPostPhys_1_1_001.tex b/SciPostPhys_1_1_001.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1969375153b5c1e70c91b2e4439c63c79b85ac82
--- /dev/null
+++ b/SciPostPhys_1_1_001.tex
@@ -0,0 +1,2019 @@
+% =========================================================================
+% SciPost LaTeX template
+% Version 1.1 (2016/05/22)
+%
+% Submissions to SciPost Journals should make use of this template.
+%
+% INSTRUCTIONS: simply look for the `TODO:' tokens and adapt your file.
+%
+% - please enable line numbers (package: lineno)
+% - you should run LaTeX twice in order for the line numbers to appear
+% =========================================================================
+
+
+% TODO: uncommente ONE of the class declarations below
+% If you are submitting a paper to SciPost Physics: uncomment next line
+\documentclass{SciPost}
+% If you are submitting a paper to SciPost Physics Lecture Notes: uncomment next line
+%\documentclass[submission,LectureNotes]{SciPost}
+
+%%%%%%%% Begin SciPost Production addition
+
+\usepackage[bitstream-charter]{mathdesign}
+
+\hypersetup{
+ colorlinks,
+ linkcolor={red!50!black},
+ citecolor={blue!50!black},
+ urlcolor={blue!80!black},
+ pdfinfo={
+ Title={Quantum quenches to the attractive one-dimensional Bose gas: exact results},
+ Author={Lorenzo Piroli, Pasquale Calabrese, Fabian H. L. Essler},
+ DOI=10.21468/SciPostPhys.1.1.001,
+ CrossMarkDomains[1]=scipost.org,
+ CrossMarkDomainExclusive=false
+ }
+}
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+\lhead{\raisebox{-1.5mm}[0pt][0pt]{\href{https://scipost.org}{\includegraphics[width=20mm]{logo_scipost_with_bgd.pdf}}}}
+\chead{}
+\rhead{\small \href{https://scipost.org/SciPostPhys.1.1.001}{SciPost Phys. 1(1), 001 (2016)}}
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+}
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+\lhead{\raisebox{-1.5mm}[0pt][0pt]{\href{https://scipost.org}{\includegraphics[width=20mm]{logo_scipost_with_bgd.pdf}}}}
+\rhead{\small \href{https://scipost.org/SciPostPhys.1.1.001}{SciPost Phys. 1(1), 001 (2016)}}
+\fancyfoot[C]{\textbf{\thepage}}
+\renewcommand{\headrulewidth}{1pt}
+}
+
+%%%%%%%% End SciPost Production addition
+
+
+
+\usepackage{cite}
+\usepackage[pdftex]{graphicx}
+
+\newcommand{\be}{\begin{equation}}
+\newcommand{\ee}{\end{equation}}
+\newcommand{\bea}{\begin{eqnarray}}
+\newcommand{\eea}{\end{eqnarray}}
+\def\HH{\mathcal H}
+
+
+\begin{document}
+
+\pagestyle{SPtitlepage}
+
+% TODO: write your article's title here.
+% The article title is centered, Large boldface, and should fit in two lines
+\begin{center}{\Large \textbf{\color{scipostdeepblue}{Quantum quenches to the attractive one-dimensional Bose gas: exact results}}}\end{center}
+
+% TODO: write the author list here. Use initials + surname format.
+% Separate subsequent authors by a comma, omit comma at the end of the list.
+% Mark the corresponding author with a superscript *.
+\begin{center}
+\textbf{L. Piroli}\textsuperscript{1*},
+\textbf{P. Calabrese}\textsuperscript{1},
+\textbf{F.H.L. Essler}\textsuperscript{2}
+\end{center}
+
+% TODO: write all affiliations here.
+% Format: institute, city, country
+\begin{center}
+{\bf 1} SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy.
+\\
+{\bf 2} The Rudolf Peierls Centre for Theoretical Physics,
+ Oxford University, Oxford, OX1 3NP, United Kingdom.
+\\[\baselineskip]
+% TODO: provide email address of corresponding author
+* lpiroli@sissa.it
+%\bigskip
+\end{center}
+
+
+%\linenumbers
+
+\section*{\color{scipostdeepblue}{Abstract}}
+{\bf
+% TODO: write your abstract here.
+We study quantum quenches to the one-dimensional Bose gas with
+attractive interactions in the case when the initial state is an ideal
+one-dimensional Bose condensate. We focus on properties of the
+stationary state reached at late times after the quench. This displays
+a finite density of multi-particle bound states, whose rapidity
+distribution is determined exactly by means of the quench action
+method. We discuss the relevance of the multi-particle bound states for
+the physical properties of the system, computing in particular the
+stationary value of the local pair correlation function $g_2$.
+}
+
+%%%%%%%% Begin SciPost Production addition
+
+%\begin{figure}[!b] % If no TOC, put copyright info at bottom of first page.
+%\noindent\rule{\textwidth}{1pt} % If no TOC
+%\vspace{-8mm} % if no TOC
+\begin{center}
+%\begin{tabular}{rlr}
+\begin{tabular}{lr}
+%&
+\begin{minipage}{0.6\textwidth}
+\raisebox{-1mm}[0pt][0pt]{\includegraphics[width=12mm]{by.eps}}
+{\small Copyright L. Piroli {\it et al}. \newline
+This work is licensed under the Creative Commons \newline
+\href{http://creativecommons.org/licenses/by/4.0/}{Attribution 4.0 International License}. \newline
+Published by the SciPost Foundation.
+}
+\end{minipage}
+&
+\begin{minipage}{0.4\textwidth}
+ \noindent\begin{minipage}{0.68\textwidth}
+{\small Received 24-05-2016 \newline Accepted 01-09-2016 \newline Published 14-09-2016
+}
+ \end{minipage}
+ \begin{minipage}{0.25\textwidth}
+ \begin{center}
+ \href{https://crossmark.crossref.org/dialog/?doi=10.21468/SciPostPhys.1.1.001&domain=pdf&date_stamp=2016-09-14}{\includegraphics[width=7mm]{CROSSMARK_BW_square_no_text.png}}\\
+ \tiny{Check for}\\
+ \tiny{updates}
+ \end{center}
+ \end{minipage}
+ \\\\
+ \small{\href{https://dx.doi.org/10.21468/SciPostPhys.1.1.001}{doi:10.21468/SciPostPhys.1.1.001}}
+\end{minipage}
+\end{tabular}
+\end{center}
+%\end{figure}
+
+%% \begin{center}
+%% \includegraphics[width=16mm]{by.eps}
+%% Copyright L. Piroli {\it et al}.
+%% This work is licensed under the Creative Commons Attribution 4.0 International License.
+
+%% Received ??-??-2016 accepted 01-09-2016 published ??-??-2016
+
+%% \end{center}
+
+%%%%%%%% End SciPost Production addition
+
+% TODO: include a table of contents (optional)
+% Guideline: if your paper is longer that 6 pages, include a TOC
+% To remove the TOC, simply cut the following block
+\vspace{10pt}
+\noindent\rule{\textwidth}{1pt}
+\tableofcontents
+%\thispagestyle{fancy}
+\noindent\rule{\textwidth}{1pt}
+\vspace{10pt}
+
+%%%%%%%% Begin SciPost Production addition
+\pagestyle{SPbulk}
+%%%%%%%% End SciPost Production addition
+
+\section{Introduction}
+
+Strongly correlated many-body quantum systems are often outside the
+range of applicability of standard perturbative methods. While being
+at the root of many interesting and sometimes surprising physical
+effects, this results in huge computational challenges, which are most
+prominent in the study of the non-equilibrium dynamics of many-body
+quantum systems.
+%
+This active field of research has attracted increasing attention over
+the last decade, also due to the enormous experimental advances in
+cold atomic physics \cite{bloch, polkovnikov, cazalilla}. Indeed,
+highly isolated many-body quantum systems can now be realised in cold
+atomic laboratories, where the high experimental control allows to
+directly probe their unitary time evolution \cite{kinoshita-06,
+ cheneau, gring,trotzky, fukuhara, langen-13, agarwal, geiger,
+ langen-15,kauf}.
+
+A simple paradigm to study the non-equilibrium dynamics of closed
+many-body quantum systems is that of a quantum quench \cite{cc-05}: a
+system is prepared in an initial state (usually the ground state of
+some Hamiltonian $H_0$) and it is subsequently time evolved with a
+local Hamiltonian $H$. In the past years, as a result of a huge
+theoretical effort (see the reviews
+ \cite{polkovnikov,efg-14,dkpr-15,ge-15,a-16,ef-16,c-16,cc-16} and
+ references therein), a robust picture has emerged: at
+ long times after the quench, and in the thermodynamic limit,
+ expectation values of {\it local} observables become stationary. For
+ a generic system, these stationary values are those of a thermal
+ Gibbs ensemble with the effective temperature being fixed by the
+energy density in the initial state \cite{rdo-08}.
+
+A different behaviour is observed for integrable quantum systems,
+where an infinite set of local conserved charges constrains the
+non-equilibrium dynamics. In this case, long times after the quench,
+local properties of the systems are captured by a generalised
+Gibbs ensemble (GGE) \cite{rdyo-07}, which is a natural extension of
+the Gibbs density matrix taking into account a complete set of local
+or quasi-local conserved charges.
+
+The initial focus was on the role played by (ultra-)local conservation
+laws in integrable quantum spin
+chains\cite{cef-11,ck-12,fe-13,bp-13a,fe-13b,kscc-13,fcec-14}, while
+more recent works have clarified the role by sets of novel,
+quasi-local charges\cite{prosen-11,pi-13,prosen-14,
+ ppsa-14,fagotti,imp-15,doyon-15,zmp-16,pv-16,fagotti-16, impz-16}.
+It has been shown recently that they have to be taken into account in
+the GGE construction in order to obtain a correct description of
+local properties of the steady state
+\cite{idwc-15,iqdb-15}. Quasi-local conservation laws and their
+relevance for the GGE have also recently been discussed in the
+framework of integrable quantum field theories
+\cite{emp-15,cardy-15}. These works have demonstrated that the problem
+of determining a complete set of local or quasi-local conserved
+charges is generally non-trivial.
+
+A different approach to calculating expectation values of local
+correlators in the stationary state was introduced in
+Ref.~\cite{ce-13}. It is the so called quench action method (QAM),
+a.k.a. representative eigenstate approach and it does not rely on the
+knowledge of the conserved charges of the system. Within this method,
+the local properties at large times are effectively described by a
+single eigenstate of the post-quench Hamiltonian. The QAM has now been
+applied to a variety of quantum quenches, from one dimensional Bose
+gases \cite{dwbc-14, dc-14, dpc-15,vwed-15, bucciantini-15, pce-16} to
+spin chains \cite{wdbf-14, bwfd-14, budapest,buda2,dmv-15} and integrable
+quantum field theories \cite{bse-14,bpc-16}, see Ref. \cite{c-16} for
+a recent review.
+
+One of the most interesting aspects of non-equilibrium dynamics in
+integrable systems is the possibility of realising non-thermal, stable
+states of matter by following the unitary time evolution after a
+quantum quench. Indeed, the steady state often exhibits properties
+that are qualitatively different from those of thermal states of
+the post-quench Hamiltonian. The QAM provides a powerful tool to
+theoretically investigate these properties in experimentally relevant
+settings.
+
+In this paper we study the quantum quench from an ideal Bose
+condensate to the Lieb-Liniger model with arbitrary attractive
+interactions. A brief account of our results was previously given in
+Ref.~\cite{pce-16}. The interest in this quench lies in its
+experimental feasibility as well as in the intriguing features of the
+stationary state, which features finite densities of multi-particle
+bound states. Our treatment, based on the quench action method, allows
+us to study their dependence on the final interaction strength and
+discuss their relevance for the physical properties of the system. In
+particular, as a meaningful example, we consider the local pair
+correlation function $g_2$, which we compute exactly.
+
+The structure of the stationary state is very different from
+the super Tonks-Girardeau gas, which is obtained by quenching the
+one-dimensional Bose gas from infinitely repulsive to infinitely
+attractive interaction \cite{abcg-05, bbgo-05,hgmd-09, mf-10, kmt-11,
+ pdc-13, th-15}. The super Tonks-Girardeau gas features no bound
+states, even though it is more strongly correlated than the infinitely
+repulsive Tonks-Girardeau gas, as has been observed experimentally
+\cite{hgmd-09}. As we argued in \cite{pce-16}, the physical properties
+of the post-quench stationary state reached in our quench protocol
+could be probed in ultracold atoms experiments, and the multi-particle
+bound states observed by the presence of different``light-cones''
+in the spreading of local correlations following a local quantum quench
+\cite{gree-12}.
+
+In this work we present a detailed derivation of the results
+previously announced in Ref.~\cite{pce-16}. The remainder of this
+manuscript is organised as follows. In section~\ref{model} we
+introduce the Lieb-Liniger model and the quench protocol that we
+consider. The quench action method is reviewed in section~\ref{QAM},
+and its application to our quench problem is detailed. In
+section~\ref{stationary_eq} the equations describing the post-quench
+stationary state are derived. Their solution is then obtained in
+section~\ref{exact_solution}, and a discussion of its properties is
+presented. In section~\ref{phys_prop} we address the calculation of
+expectation values of certain local operators on the post-quench
+stationary state, and we explicitly compute the local pair correlation
+function $g_2$. Finally, our conclusions are presented in
+section~\ref{conclusions}. For the sake of clarity, some technical
+aspects of our work are consigned to several appendices.
+
+
+\section{The Lieb-Liniger model}
+\label{model}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{The Hamiltonian and the eigenstates}
+\label{eigenstates}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+We consider the Lieb-Liniger model \cite{lieb}, consisting of $N$
+interacting bosons on a one-dimensional ring of circumference $L$. The
+Hamiltonian reads
+\begin{equation}
+H^N_{LL}=-\frac{\hbar^2}{2m}\sum_{j=1}^{N}\frac{\partial^2}{\partial x_{j}^2}+2c\sum_{j<k}\delta(x_j-x_k),
+\label{hamiltonian}
+\end{equation}
+where $m$ is the mass of the bosons, and $c=-\hbar^2/ma_{1D}$ is the
+interaction strength. Here $a_{1D}$ is the 1D effective scattering
+length \cite{olshanii-98} which can be tuned via Feshbach resonances
+\cite{iasm-98}. In the following we fix $\hbar=2m=1$. The second
+quantized form of the Hamiltonian is
+\begin{equation}
+H_{LL}=\int_{0}^{L}\mathrm{d}x \Big\{\partial_x\Psi^{\dagger}(x)\partial_x\Psi(x)+c\Psi^{\dagger}(x)\Psi^{\dagger}(x)\Psi(x)\Psi(x)\Big\},
+\label{hamiltonian_2}
+\end{equation}
+where $\Psi^{\dagger}$, $\Psi$ are complex bosonic fields satisfying
+$[\Psi(x),\Psi^{\dagger}(y)]=\delta(x-y)$.
+
+The Hamiltonian (\ref{hamiltonian}) can be exactly diagonalised for
+all values of $c$ using the Bethe ansatz method
+\cite{lieb,korepin}. Throughout this work we will consider the
+attractive regime $c<0$ and use notations
+$\overline{c}=-c>0$. We furthermore define a dimensionless coupling
+constant by
+\begin{equation}
+\gamma=\frac{\overline{c}}{D}\ ,\quad D=\frac{N}{L}.
+\end{equation}
+
+A general $N$-particle energy eigenstate is parametrized by a set of
+$N$ complex rapidities $\{\lambda_j\}_{j=1}^{N}$, satisfying the
+following system of Bethe equations
+\begin{equation}
+e^{-i\lambda_jL}=\prod_{k\neq j\atop k=1}^{N}\frac{\lambda_k-\lambda_j-i\overline{c}}{\lambda_k-\lambda_j+i\overline{c}}\ ,\quad j=1,\ldots, N\ .
+\label{bethe_eq}
+\end{equation}
+The wave function of the eigenstate corresponding to the set of rapidities
+$\{\lambda_j\}_{j=1}^{N}$ is then
+\begin{equation}
+\psi_N(x_1,\ldots,x_N|\{\lambda_j\}_{j=1}^N)=\frac{1}{\sqrt{N}}\sum_{P\in \mathcal{S}_N}e^{i\sum_{j}x_j\lambda_{P_j}} \prod_{j>k}\frac{\lambda_{P_j}-\lambda_{P_k}+i\overline{c}\mathrm{sgn}(x_j-x_k)}{\lambda_{P_j}-\lambda_{P_k}},
+\end{equation}
+where the sum is over all the permutations of the rapidities. Eqns
+(\ref{bethe_eq}) can be rewritten in logarithmic form as
+\begin{equation}
+\lambda_jL-2\sum_{k=1}^{N}\arctan\left(\frac{\lambda_j-\lambda_k}{\overline{c}}\right)=2\pi I_j\ ,\quad j=1,\ldots, N\ ,
+\label{bethe_log}
+\end{equation}
+where the quantum numbers $\{I_j\}_{j=1}^{N}$ are integer (half-odd
+integer) for $N$ odd (even).
+
+In the attractive regime the solutions of (\ref{bethe_log}) organize themselves into mutually disjoint patterns in the complex rapidity plane called ``strings'' \cite{takahashi, cc-07}. For a given $N$ particle state, we indicate with $\mathcal{N}_s$ the total number of strings and with $N_j$ the number of $j$-strings, i.e. the strings containing $j$ particles ($1\leq j\leq N$) so that
+\begin{equation}
+N=\sum_{j}jN_j,\qquad \mathcal{N}_s=\sum_{j}N_j.
+\end{equation}
+The rapidities within a single $j$-string are parametrized as\cite{mg-64}
+\begin{equation}
+\lambda^{j,a}_{\alpha}=\lambda_{\alpha}^{j}+\frac{i\overline{c}}{2}(j+1-2a)+i\delta^{j,a}_{\alpha} ,\quad a=1,\ldots, j ,
+\label{structure}
+\end{equation}
+where $a$ labels the individual rapidities within the $j$-string, while
+$\alpha$ labels different strings of length $j$. Here
+$\lambda_{\alpha}^j$ is a real number called the string centre. The
+structure (\ref{structure}) is common to many integrable systems and
+within the so called string hypothesis \cite{takahashi, thacker} the
+deviations from a perfect string $\delta^{j,a}_{\alpha}$ are assumed
+to be exponentially vanishing with the system size $L$ (see
+Refs.~\cite{sakmann, sykes} for a numerical study of such deviations
+in the Lieb-Liniger model). A $j$-string can be seen to correspond to
+a bound state of $j$ bosons: indeed, one can show that the Bethe
+ansatz wave function decays exponentially with respect to the distance
+between any two particles in the bound state and the $j$ bosons can be
+thought as clustered together.
+
+Even though some cases are known where states violating the string
+hypothesis are present \cite{vladimirov, essler-92, ilakovac, fujita,
+ hagemans}, it is widely believed that their contribution to
+physically relevant quantities is vanishing in the thermodynamic
+limit. We will then always assume the deviations
+$\delta^{j,a}_{\alpha}$ to be exponentially small in $L$ and neglect
+them except when explicitly said otherwise.
+
+From (\ref{bethe_log}), (\ref{structure}) a system of equations for
+the string centres $\lambda^j_{\alpha}$ is obtained \cite{cc-07}
+\begin{equation}
+j\lambda_{\alpha}^{j}L-\sum_{(k,\beta)}\Phi_{jk}(\lambda^{j}_{\alpha}-\lambda_{\beta}^{k})=2\pi I^{j}_{\alpha}\ ,
+\label{BGT}
+\end{equation}
+where
+\bea
+\Phi_{jk}(\lambda)&=&(1-\delta_{jk})\phi_{|j-k|}(\lambda)+2\phi_{|j-k|+2}(\lambda)+\ldots+2\phi_{j+k-2}(\lambda)+\phi_{j+k}(\lambda)\ ,
+\\
+\phi_j(\lambda)&=&2\arctan\left(\frac{2\lambda}{j\overline{c}}\right)\ ,
+\eea
+and where $I^j_{\alpha}$ are integer (half-odd integer) for $N$ odd (even).
+Eqns (\ref{BGT}) are called Bethe-Takahashi equations
+\cite{takahashi,gaudin}. The momentum and the energy of a general
+eigenstate are then given by
+\be
+K=\sum_{(j,\alpha)} j \lambda^j_{\alpha}\ ,\qquad E = \sum_{(j,\alpha)} j (\lambda^j_{\alpha})^2 - \frac{\bar c^2}{12} j(j^2 - 1).
+\label{momentum_energy}
+\ee
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{The thermodynamic limit}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In the repulsive case the thermodynamic limit
+\begin{equation}
+N,L\to \infty\ ,\quad
+D=\frac{N}{L}\ {\rm fixed},
+\end{equation}
+was first considered in Ref.~\cite{yy-69}, and it is well studied in
+the literature. In the attractive case, the absolute value of the
+ground state energy in not extensive, but instead grows as $N^3$
+\cite{mg-64,cd-75}. While ground state correlation functions can be
+studied in the zero density limit, namely $N$ fixed, $L\to \infty$
+\cite{cc-07}, it was argued that the model does not have a proper
+thermodynamic limit in thermal equilibrium
+\cite{cd-75,takahashi}. Crucially, in the quench protocol we are
+considering, the energy is fixed by the initial state and the limit of
+an infinite number of particles at fixed density presents no problem.
+
+As the systems size $L$ grows, the centres of the strings associated
+with an energy eigenstate become a dense set on the real line and in the
+thermodynamic limit are described by smooth distribution function.
+In complete analogy with the standard finite-temperature formalism
+\cite{takahashi} we introduce the distribution function
+$\{\rho_n(\lambda)\}_{n=1}^{\infty}$ describing the centres of $n$
+strings, and the distribution function of holes
+$\{\rho^h_n(\lambda)\}_{n=1}^{\infty}$. We recall that
+$\rho^h_n(\lambda)$ describes the distribution of unoccupied states for
+the centres of $n$-particle strings, and is analogous to the
+distribution of holes in the case of ideal Fermi gases at finite
+temperature. Following Takahashi~\cite{takahashi} we introduce
+\bea
+\eta_{n}(\lambda)&=&\frac{\rho_{n}^{h}(\lambda)}{\rho_{n}(\lambda)},\label{eq:eta}\\
+\rho^{t}_{n}(\lambda)&=&\rho_n(\lambda)+\rho_{n}^h(\lambda). \label{rho_tot}
+\eea
+In the thermodynamic limit the Bethe-Takahashi equations (\ref{BGT})
+reduce to an infinite set of coupled, non-linear integral equations
+\begin{equation}
+\frac{n}{2\pi}-\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}\lambda'a_{nm}(\lambda-\lambda')\rho_{m}(\lambda')=\rho_{n}(\lambda)(1+\eta_{n}(\lambda)).
+\label{coupled}
+\end{equation}
+where
+\bea
+a_{nm}(\lambda)&=&(1-\delta_{nm})a_{|n-m|}(\lambda)+2a_{|n-m|+2}(\lambda)+\ldots+2a_{n+m-2}(\lambda)+a_{n+m}(\lambda)\ ,
+\label{aa_function}\\
+ a_{n}(\lambda)&=&\frac{1}{2\pi}\frac{\mathrm{d}}{\mathrm{d}\lambda}\phi_{n}(\lambda)=\frac{2}{\pi n \overline{c}}\frac{1}{1+\left(\frac{2\lambda}{n \overline{c}}\right)^2}\ .
+\label{a_function}
+\eea
+In the thermodynamic limit the energy and momentum per volume are
+given by
+\bea
+k[\{\rho_n\}]=\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}{\rm d}\lambda\ \rho_n(\lambda)n\lambda,\qquad e[\{\rho_n\}]=\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}{\rm d}\lambda\ \rho_n(\lambda)\varepsilon_n(\lambda),
+\label{therm_momentum_energy}
+\eea
+where
+\be
+\varepsilon_{n}(\lambda)=n \lambda^2 - \frac{\bar c^2}{12} n(n^2 - 1).
+\label{eq:epsilon}
+\ee
+Finally, it is also useful to define the densities $D_n$ and energy densities
+$e_n$ of particles forming $n$-strings
+\be
+D_n= n\int_{-\infty}^{\infty}{\rm d}\lambda\ \rho_n(\lambda),\qquad
+e_n=\int_{-\infty}^{\infty}{\rm d}\lambda\ \rho_n(\lambda)\varepsilon_n(\lambda).
+\label{density_per_string}
+\ee
+The total density and energy per volume are then additive
+\be
+D=\sum_{n=1}^{\infty}D_n,\qquad e=\sum_{n=1}^{\infty}e_n.
+\label{Dtot}
+\ee
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{The quench protocol}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+We consider a quantum quench in which the system is initially prepared
+in the BEC state, i.e. the ground state of (\ref{hamiltonian}) with
+$c=0$, and the subsequent unitary time evolution is governed by the
+Hamiltonian (\ref{hamiltonian}) with $c=-\overline{c}<0$. The same
+initial state was considered for quenches to the repulsive Bose
+gas in Refs~\cite{kscc-13, dwbc-14, dc-14, kcc-14, grd-10, zwkg-15},
+while different initial conditions were considered in
+Refs~\cite{m-13,mossel-c-12, ia-12, csc-13, ga-14,sc-14, mckc-14,
+ fgkt-15, goldstein-15, bucciantini-15, cgfb-14,gfcb-16}.
+
+As we mentioned before, the energy after the quench is conserved and
+is most easily computed in the initial state $|\psi(0)\rangle=|{\rm BEC}\rangle$ as
+\be
+\langle {\rm BEC}|H_{LL}|{\rm BEC}\rangle= -\overline{c}\langle{\rm BEC}|\int_0^{L}{\rm d}x\ \Psi^{\dagger}(x)\Psi^{\dagger}(x)\Psi(x)\Psi(x)|{\rm BEC}\rangle.
+\ee
+The expectation value on the r.h.s. can then be easily computed using
+Wick's theorem. In the thermodynamic limit we have
+\be
+\frac{E}{L}=-\overline{c}D^2=-\gamma D^3.
+\label{energy}
+\ee
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{The quench action method}
+\label{QAM}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{General considerations}
+Consider the post-quench time evolution of the expectation value of a general
+operator ${O}$. For a generic system it can be written as
+\be
+\langle \psi(t) | {O} | \psi(t) \rangle= \sum_{\mu, \nu} \langle
+\psi(0)|\mu\rangle \langle \mu|{O}|\nu\rangle\langle \nu | \psi(0)\rangle
+e^{i (E_\mu-E_\nu)t},
+\label{ds}
+\ee
+where $\{|\mu\rangle\}$ denotes an orthonormal basis of eigenstates of
+the post-quench Hamiltonian. In Ref.~\cite{ce-13} it was argued that in
+integrable systems a major simplification occurs if one is interested
+in the time evolution of the expectation values of {\it local}
+operators $\mathcal{O}$ in the thermodynamic limit. In particular, the
+double sum in the spectral representation (\ref{ds}) can be
+replaced by a single sum over particle-hole excitations over
+a \emph{representative eigenstate} $|\rho_{sp}\rangle$. In
+particular, we have
+\be
+{\rm lim}_{\rm th}\langle \psi(t) | \mathcal{O} | \psi(t) \rangle= \frac{1}{2}\sum_{{\rm {\bf e}}}\left(e^{-\delta s_{{\rm {\bf e}}} - i\delta \omega_{{\rm {\bf e}}} t}\langle \rho_{sp}|\mathcal{O}|\rho_{sp},{\rm {\bf e}}\rangle + e^{-\delta s^{\ast}_{{\rm {\bf e}}} + i\delta \omega_{{\rm {\bf e}}} t}\langle \rho_{sp},{\rm {\bf e}}|\mathcal{O}|\rho_{sp}\rangle \right),
+\label{time_ev}
+\ee
+where we have indicated with ${\rm lim}_{\rm th}$ the thermodynamic
+limit $N,L\to \infty$, keeping the density $D=N/L$ fixed. Here {\bf e}
+denotes a generic excitation over the representative state
+$|\rho_{sp}\rangle$. Finally we have
+\be
+\delta s_{{\rm {\bf e}}}=-\ln\frac{\langle \rho_{sp}, {\rm {\bf e}}|\psi(0)\rangle}{\langle \rho_{sp}|\psi(0)\rangle}, \qquad \delta\omega_{{\rm {\bf e}}}=\omega[\rho_{sp},{\rm {\bf e}}]-\omega[\rho_{sp}],
+\ee
+where $\omega[\rho_{sp}]$, $\omega[\rho_{sp},{\rm {\bf e}}]$ are the
+energies of $|\rho_{sp}\rangle$ and $|\rho_{sp}, {\rm {\bf
+ e}}\rangle$ respectively. The representative eigenstate (or
+``saddle-point state'') $|\rho_{sp}\rangle$ is described in the
+thermodynamic limit by two sets of distribution functions
+$\{\rho_{n}(\lambda)\}_{n}$, $\{\rho^{h}_{n}(\lambda)\}_{n}$. In
+Ref.~\cite{ce-13} it was argued that these are selected by the
+saddle-point condition
+\be
+\frac{\partial S_{QA}[\rho]}{\partial \rho_n(\lambda)} \Big|_{\rho=\rho_{sp}}=0, \qquad n\geq 1,
+\label{oTBA1}
+\ee
+where $S_{QA}[\rho]$ is the so-called quench action
+\be
+S_{QA}[\rho]=2S[\rho]-S_{YY}[\rho].
+\label{SQA}
+\ee
+Here $\rho$ is the set of distribution functions corresponding to
+a general macro-state, $S[\rho]$ gives the thermodynamically leading
+part of the logarithm of the overlap
+\be
+S[\rho]=-{\rm lim}_{\rm th}{\rm Re}\ln\langle \psi(0)|\rho\rangle,
+\label{therm_overlap}
+\ee
+and $S_{YY}$ is the Yang-Yang entropy. As we will see in section
+ \ref{overlap_bec}, we will only have to consider parity-invariant
+ Bethe states, namely eigenstates of the Hamiltonian
+ (\ref{hamiltonian}) characterised by sets of rapidities satisfying
+ $\{\lambda_j\}_{j=1}^{N}=\{-\lambda_j\}_{j=1}^{N}$. Restricting to
+ the sector of the Hilbert space of parity invariant Bethe states, the
+ Yang-Yang entropy reads
+\be
+\frac{S_{YY}[\rho]}{L}= \frac{1}{2}\sum_{n=1}^\infty \int_{-\infty}^{\infty} d\lambda [\rho_n \ln (1+\eta_n)+ \rho_n^h \ln (1+\eta_n^{-1})].
+\label{yang}
+\ee
+We note the global pre-factor $1/2$. From Eq. (\ref{time_ev}) it
+follows that the saddle-point state $|\rho_{sp}\rangle$ can be seen
+as the effective stationary state reached by the system at long
+times. Indeed, if $\mathcal{O}$ is a local operator,
+Eq. (\ref{time_ev}) gives
+\begin{equation}
+\lim_{t\to\infty}{\rm lim}_{\rm th}\langle \psi(t) | \mathcal{O} | \psi(t) \rangle =\langle \rho_{sp}|\mathcal{O}|\rho_{sp}\rangle .
+\end{equation}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Overlaps with the BEC state}
+\label{overlap_bec}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The main difficulty in applying the quench action method to a generic
+quantum quench problems is the computation of the overlaps
+$\langle\psi(0)| \rho\rangle$ between the initial state and eigenstates of the
+post-quench Hamiltonian. At present this problem has been solved only
+in a small number of special cases \cite{ce-13,mossel-10, pozsgay-14,
+ amsterdam_overlaps, brockmann, pc-14, hst-15,
+ mazza-15,fz-16,cl-14}.
+
+A conjecture for the overlaps between the BEC state and the Bethe
+states in the Lieb-Liniger model first appeared in Ref.~\cite{dwbc-14}
+and it was then rigorously proven, for arbitrary sign of the particle
+interaction strength, in Ref.~\cite{brockmann}. As we have already
+mentioned, the overlap is non-vanishing only for parity invariant
+Bethe states, namely eigenstates characterised by sets of rapidities
+satisfying $\{\lambda_j\}_{j=1}^{N}=\{-\lambda_j\}_{j=1}^{N}$
+\cite{amsterdam_overlaps}. The formula reads
+\be
+\langle \{\lambda_{j}\}_{j=1}^{N/2}\cup \{-\lambda_{j}\}_{j=1}^{N/2} |{\rm BEC}\rangle=\frac{\sqrt{(cL)^{-N}N!}}{\prod_{j=1}^{N/2}\frac{\lambda_j}{c}\sqrt{\frac{\lambda_j^2}{c^2}+\frac{1}{4}}}\frac{\mathrm{det}^{N/2}_{j,k,=1}G_{jk}^{Q}}{\sqrt{\mathrm{det}^{N}_{j,k,=1}G_{jk}}},
+\label{overlap}
+\ee
+where
+\be
+G_{jk}=\delta_{jk}\left[L+\sum_{l=1}^{N}K(\lambda_j-\lambda_l)\right]-K(\lambda_j-\lambda_k),
+\ee
+\be
+G^Q_{jk}=\delta_{jk}\left[L+\sum_{l=1}^{N/2}K^Q(\lambda_j,\lambda_l)\right]-K^Q(\lambda_j,\lambda_k),
+\ee
+\be
+K^Q(\lambda,\mu)=K(\lambda-\mu)+K(\lambda+\mu),\qquad K(\lambda)=\frac{2c}{\lambda^2+c^2}.
+\ee
+The extensive part of the logarithm of the overlap (\ref{overlap}) was
+computed in Ref.~\cite{dwbc-14} in the repulsive regime. A key
+observation was that the ratio of the determinants is non-extensive, i.e.
+\be
+{\rm lim}_{\rm th} \frac{\mathrm{det}^{N/2}_{j,k,=1}G_{jk}^{Q}}{\sqrt{\mathrm{det}^{N}_{j,k,=1}G_{jk}}}=\mathcal{O}(1).
+\ee
+
+In the attractive regime additional technical difficulties arise,
+because matrix elements of the Gaudin-like matrices $G_{jk}$,
+$G^{Q}_{jk}$ can exhibit singularities when the Bethe state contains
+bound states \cite{cl-14}. This is analogous to the situation
+encountered for a quench from the N\'{e}el state to the gapped XXZ
+model \cite{wdbf-14, bwfd-14,budapest,buda2}. In particular, one can
+see that the kernel $K(\mu-\nu)$ diverges as
+$1/(\delta_{\alpha}^{n,a}-\delta_{\alpha}^{n,a+1})$ for two
+``neighboring'' rapidities in the same string
+$\mu=\lambda_{\alpha}^{n,a}$, $\nu=\lambda_{\alpha}^{n,a+1}$, or when
+rapidities from different strings approach one another in the
+thermodynamic limit, $\mu\to\lambda+ic$.
+
+These kinds of singularities are present in the determinants of both
+$G^{Q}_{jk}$ and $G_{jk}$. It was argued in Refs~\cite{wdbf-14,
+ bwfd-14,cl-14} that they cancel one another in the expression for
+the overlap. As was noted in Refs.~\cite{wdbf-14,bwfd-14,cl-14}, no
+other singularities arise as long as one considers the overlap between
+the BEC state and a Bethe state without zero-momentum $n$-strings,
+(strings centred at $\lambda=0$). Concomitantly the ratio of the
+determinants in (\ref{overlap}) is expected to give a sub-leading
+contribution in the thermodynamic limit, and can be dropped. The
+leading term in the logarithm of the overlaps can then be easily
+computed along the lines of Refs.~\cite{wdbf-14, bwfd-14}
+\be
+\ln \langle\rho|{\rm BEC}\rangle=-\frac{LD}{2}\left(\ln\gamma+1\right)+\frac{L}{2}\sum_{m=1}^{\infty}\int_{0}^{\infty}d\lambda \rho_{n}(\lambda)\ln W_{n}(\lambda),
+\label{leading}
+\ee
+where
+\be
+W_n(\lambda)=\frac{1}{\frac{\lambda^2}{\overline{c}^2}\left(\frac{\lambda^2}{\overline{c}^2}+\frac{n^2}{4}\right)\prod_{j=1}^{n-1}\left(\frac{\lambda^2}{\overline{c}^2}+\frac{j^2}{4}\right)^2}.
+\label{w_n}
+\ee
+In the case where zero-momentum $n$-strings are present, a more
+careful analysis is required in order to extract the leading term of
+the overlap (\ref{overlap}) \cite{cl-14,ac-15}. This is reported in Appendix~\ref{app_overlap}. The upshot of this analysis is that (\ref{leading})
+gives the correct leading behaviour of the overlap even in the
+presence of zero-momentum $n$-strings.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Stationary state}
+\label{stationary_eq}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Saddle point equations}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+As noted before, the stationary state is characterized by two sets
+of distribution functions $\{\rho_n(\lambda)\}_n$, $\{\rho^h_n(\lambda)\}_n$,
+which fulfil two infinite systems of coupled, non-linear integral
+equations.
+The first of these is the thermodynamic version of the Bethe-Takahashi
+equations (\ref{coupled}). The second set is derived from the
+saddle-point condition of the quench action (\ref{oTBA1}), and the
+resulting equations are sometimes called the overlap thermodynamic
+Bethe ansatz equations (oTBA equations). Their derivation follows
+Refs~\cite{wdbf-14,bwfd-14,budapest,buda2}. In order to fix the density
+$D=N/L$ we add the following term to the quench action (\ref{SQA})
+\be
+-hL\left(\sum_{m=1}^{\infty}m\int_{-\infty}^{\infty}d\lambda\rho_{m}(\lambda)-D\right).
+\label{density_condition}
+\ee
+As discussed in the previous section, $S[\rho]$ in (\ref{SQA}) can be
+written as
+\be
+S[\rho]=\frac{LD}{2}\left(\ln\gamma+1\right)-\frac{L}{2}\sum_{m=1}^{\infty}\int_{0}^{\infty}d\lambda \rho_{n}(\lambda)\ln W_{n}(\lambda)\ ,
+\label{s_term}
+\ee
+where $W_n(\lambda)$ is given in (\ref{w_n}). Using (\ref{s_term}),
+(\ref{yang}), and (\ref{density_condition}) one can straightforwardly
+extremize the quench action (\ref{SQA}) and arrive
+at the following set of oTBA equations
+\begin{equation}
+\ln\eta_{n}(\lambda)=-2hn-\ln W_{n}(\lambda)+\sum_{m=1}^{\infty}a_{nm}\ast \ln\left(1+\eta_{m}^{-1}\right)(\lambda),\qquad n\geq 1\ .
+\label{coupled2}
+\end{equation}
+Here $a_{nm}$ are defined in (\ref{aa_function}), and we have used the notation
+$f\ast g(\lambda)$ to indicate the convolution between two functions
+\be
+f\ast g(\lambda)=\int_{-\infty}^{\infty}{\rm d}\mu \ f(\lambda-\mu)g(\mu).
+\label{convolution}
+\ee
+Eqns (\ref{coupled2}) determine the functions $\eta_{n}(\lambda)$ and,
+together with Eqns (\ref{coupled}) completely fix the distribution functions
+$\{\rho_n(\lambda)\}_n$, $\{\rho^{h}_n(\lambda)\}_n$ characterising
+the stationary state.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Tri-diagonal form of the oTBA equations}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Following standard manipulations of equilibrium TBA equations
+\cite{takahashi}, we may re-cast the oTBA equations (\ref{coupled2})
+in the form
+\bea
+\ln\eta_n(\lambda)=d(\lambda)+s\ast\left[\ln(1+\eta_{n-1})+\ln(1+\eta_{n+1})\right](\lambda)\ ,\qquad n\geq 1\ .
+\label{finale_gtba}
+\eea
+Here we have defined $\eta_0(\lambda)=0$ and
+\bea
+s(\lambda)=\frac{1}{2\overline{c}\cosh\left(\frac{\pi\lambda}{\overline{c}}\right)},\label{kernel}\\
+d(\lambda)=\ln\left[\tanh^2\left(\frac{\pi\lambda}{2\overline{c}}\right)\right]. \label{a_driving}
+\eea
+The calculations leading to Eqns (\ref{finale_gtba}) are summarized in Appendix~\ref{app_tridiag}.
+The thermodynamic form of the Bethe-Takahashi equations
+(\ref{coupled}) can be similarly rewritten. Since we do not make
+explicit use of them in the following, we relegate their derivation to
+Appendix~\ref{app_tridiag}.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Asymptotic relations}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Eqns~(\ref{finale_gtba}) do not fix $\{\eta_{n}(\lambda)\}_n$ of
+Eqns~(\ref{coupled2}), because they do not contain the chemical
+potential $h$. In order to recover the (unique) solution of
+Eqns~(\ref{coupled2}), it is then necessary
+to combine Eqns~(\ref{finale_gtba}) with a condition on the asymptotic
+behaviour of $\eta_{n}(\lambda)$ for large $n$. In our case one can
+derive from (\ref{coupled2}) the following relation, which holds
+asymptotically for $n\to\infty$
+\be
+\ln\eta_{n+1}(\lambda)\simeq -2h+a_1\ast \ln\eta_n(\lambda)+\ln\left[\frac{\lambda}{\overline{c}}\left(\frac{\lambda^2}{\overline{c}^2}+\frac{1}{4}\right)\right].
+\label{difference}
+\ee
+Here $a_1(\lambda)$ is given in (\ref{a_function}) (for $n=1$).
+The derivation of Eqn~(\ref{difference}) is reported in
+Appendix~\ref{app_asymptotic}. The set of equations (\ref{finale_gtba}), with
+the additional constraint given by Eqn~(\ref{difference}), is now
+equivalent to Eqns~(\ref{coupled2}).
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Rapidity distribution functions for the stationary state}
+\label{exact_solution}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Numerical analysis}
+\label{numerics}
+
+Eqns (\ref{coupled}), (\ref{coupled2}) can be truncated
+to obtain a finite system of integral equations, which are defined on
+the real line $\lambda\in(-\infty,\infty)$. One can then numerically
+solve this finite system either by introducing a cut-off for large
+$\lambda$, or by mapping the equations onto a finite
+interval. Following the latter approach, we define
+\begin{equation}
+\chi_{n}(\lambda)=\ln \left(\frac{\eta_{n}(\lambda)\tau^{2n}}{q_{n}(\lambda)}\right)\,,
+\label{new_functions}
+\end{equation}
+where $q_{n}(\lambda)$ is given by
+\be
+q_n(\lambda)=\frac{1}{W_n(\lambda)}=\frac{\lambda^2}{\overline{c}^2}\left(\frac{\lambda^2}{\overline{c}^2}+\left(\frac{n}{2}\right)^2\right)\prod_{l=1}^{n-1}\left[\frac{\lambda^2}{\overline{c}^2}+\left(\frac{l}{2}\right)^2\right]^2\ .
+\label{poly}
+\ee
+Finally, we have defined
+\be
+\tau=e^h,
+\label{tau}
+\ee
+$h$ being the Lagrange multiplier appearing in (\ref{coupled2}).
+The functions $\chi_n(\lambda)$ satisfy the following system of equations
+\bea
+\chi_{n}(\lambda)&=&\sum_{m=1}^{\infty}a_{nm}\ast \ln\left(1+\frac{\tau^{2m}}{q_m(\lambda)}e^{-\chi_m(\lambda)}\right)=\nonumber \\
+&=&\sum_{m=1}^{\infty}\int_{0}^{+\infty}\mathrm{d}\ \mu\ (a_{nm}(\lambda-\mu)+a_{nm}(\lambda+\mu))\ln\left(1+\frac{\tau^{2m}}{q_m(\mu)}e^{-\chi_m(\mu)}\right),
+\label{modified}
+\eea
+where $a_{nm}(\lambda)$ are defined in (\ref{aa_function}). We then
+change variables
+\be
+\frac{\lambda(x)}{\overline{c}}=\frac{1-x}{1+x}\ ,
+\label{map}
+\ee
+which maps the interval $(0,\infty)$ onto $(-1,1)$. Since the distributions $\chi_{n}(\lambda)$ are symmetric w.r.t. $0$, they can be described by functions with domain $(0,\infty)$. Using the map (\ref{map}) they become functions $\chi_{n}(x)$ with domain $(-1,1)$. The set of equations (\ref{modified}) becomes
+\be
+\chi_{n}(x)=2\sum_{m=1}^{\infty}\int_{-1}^{1}\ \mathrm{d}y \frac{1}{(1+y)^2}\mathcal{A}_{nm}(x,y)\ln\left(1+\frac{\tau^{2m}}{q_m(y)}e^{-\chi_m(y)}\right)\ ,
+\label{compact1}
+\ee
+where
+\be
+\mathcal{A}_{nm}(x,y)=\overline{c}\left[a_{nm}\bigg(\lambda(x)-\lambda(y)\bigg)
++a_{nm}\bigg(\lambda(x)+\lambda(y)\bigg)\right]\,.
+\label{new_function_2}
+\ee
+The thermodynamic Bethe-Takahashi equations (\ref{coupled}) can be
+similarly recast in the form
+\be
+\Theta_{n}(x)=\frac{n}{2\pi}-2\sum_{m=1}^{\infty}\int_{-1}^{1}\frac{\mathrm{d}y}{(1+y)^2}\frac{\mathcal{A}_{nm}(x,y)}{1+\eta_{m}(y)}\Theta_{m}(y),
+\label{compact2}
+\ee
+where $\Theta(x)=\rho^t_n\big(\lambda(x)\big)$, with $\lambda(x)$
+defined in Eq. (\ref{map}). The infinite systems (\ref{compact1}) and
+(\ref{compact2}), defined on the interval $(-1,1)$, can then be
+truncated and solved numerically for the functions $\chi_n(x)$ and
+$\Theta_n(x)$, for example using the Gaussian quadrature method. The
+functions $\eta_n(\lambda)$ are recovered from (\ref{new_functions})
+and (\ref{map}), while the particle and hole distributions
+$\rho_n(\lambda)$, $\rho_n^h(\lambda)$ are obtained from the
+knowledge of $\eta_n(\lambda)$ and $\rho_n^t(\lambda)$.
+
+As $\gamma$ decreases, we found that an increasing number of equations
+has to be kept when truncating the infinite systems (\ref{compact1}),
+(\ref{compact2}) in order to obtain an accurate numerical solution. As
+we will see in section \ref{physical_discussions}, this is due to the
+fact that, as $\gamma\to 0$, bound states with higher number of
+particles are formed and the corresponding distribution functions
+$\rho_n(\lambda)$, $\eta_n(\lambda)$ cannot be neglected in
+(\ref{coupled}), (\ref{coupled2}). As an example, our numerical
+solution for $\gamma=0.25$, and $\gamma=2.5$ is shown in
+Fig.~\ref{distributions}, where we also provide a comparison with the
+analytical solution discussed in section~\ref{analytical}.
+
+Two non-trivial checks for our numerical solution are available.
+The first is given by Eq.~(\ref{energy}), i.e. the solution has to
+satisfy the sum rule
+\be
+\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}{\rm d}\lambda\rho_n(\lambda)\varepsilon_n(\lambda)=-\gamma D^3,
+\label{check1}
+\ee
+where $\varepsilon_n(\lambda)$ is defined in
+Eqn (\ref{eq:epsilon}). The second non-trivial check was suggested in
+Refs~\cite{budapest,buda2} (see also Ref.~\cite{bwfd-14}), and is
+based on the observation that the action (\ref{SQA}) has to be equal to
+zero when evaluated on the saddle point solution,
+i.e. $S_{QA}[\rho_{sp}]=0$, or
+\be
+2S[\rho_{sp}]=S_{YY}[\rho_{sp}],
+\label{check2}
+\ee
+where $S[\rho]$ and $S_{YY}[\rho]$ are defined respectively in
+(\ref{s_term}) and (\ref{yang}). Both (\ref{check1}) and
+(\ref{check2}) are satisfied by our numerical solutions within a
+relative numerical error $\epsilon \lesssim 10^{-4}$ for all
+numerically accessible values of $h$. As a final check we have
+verified that our numerical solution satisfies, within numerical errors,
+\begin{equation}
+\gamma=\frac{1}{\tau}\ ,
+\label{fact}
+\end{equation}
+where $\tau$ is defined in (\ref{tau}) and $\gamma=\bar{c}/D$ is computed from
+the distribution functions using (\ref{Dtot}). Relation (\ref{fact})
+is equivalent to that found in the repulsive case \cite{dwbc-14}.
+
+\begin{figure}
+\centering
+%\includegraphics[scale=0.93]{fig1.pdf}
+\includegraphics[width=\textwidth]{fig1.pdf}
+\caption{Rapidity distribution functions $\rho_n(\lambda)$ and $(2\pi/
+n)\rho_n^{h}(\lambda)$ for $n$-string solutions with $n\leq 4$. The
+final value of the interaction is chosen as
+($a$) $\gamma=0.25$ and ($b$) $\gamma=2.5$. The dots correspond to
+the numerical solution discussed in section~\ref{numerics}, while
+solid lines correspond to the analytical solution presented in
+section \ref{analytical}. The functions are shown for $\lambda>0$
+(being symmetric with respect to $\lambda=0$) and have been rescaled
+for presentational purposes. Note that the rescaling factors for the hole
+distributions are determined by their asymptotic values,
+$\rho_n^h(\lambda)\to n/2\pi$ as $\lambda\to \infty$.}
+\label{distributions}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Perturbative expansion}
+\label{perturbative_sec}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Following Ref.~\cite{dwbc-14} we now turn to a ``perturbative''
+analysis of Eqns (\ref{coupled2}). This will provide us with another
+non-trivial check on the validity of the analytical solution presented
+in section \ref{analytical}.
+Defining $\varphi_n(\lambda)=1/\eta_n(\lambda)$ and using (\ref{tau}),
+we can rewrite (\ref{coupled2}) in the form
+\be
+\ln \varphi_n(\lambda)=\ln(\tau^{2n})+\ln W_n(\lambda)-\sum_{m=1}^{\infty}a_{nm}\ast\ln(1+\varphi_m)(\lambda),
+\label{perturbative}
+\ee
+where $W_n(\lambda)$ is given in (\ref{w_n}). We now expand the
+functions $\varphi_n(\lambda)$ as power series in $\tau$
+\be
+\varphi_n(\lambda)=\sum_{k=0}^{\infty}\varphi^{(k)}_n(\lambda)\tau^k.
+\ee
+From (\ref{perturbative}) one readily sees that $\varphi_n(\lambda)=\mathcal{O}(\tau^{2n})$, i.e.
+\bea
+\varphi^{(k)}_n(\lambda)=0,\quad k=0,\ldots, 2n-1,\\
+\varphi^{(2n)}_n(\lambda)=\frac{1}{\frac{\lambda^2}{\overline{c}^2}\left(\frac{\lambda^2}{\overline{c}^2}+\frac{n^2}{4}\right)\prod_{j=1}^{n-1}\left(\frac{\lambda^2}{\overline{c}^2}+\frac{j^2}{4}\right)^2}.
+\label{coefficient}
+\eea
+
+Using (\ref{coefficient}) as a starting point we can now solve
+Eqns (\ref{perturbative}) by iteration. The calculations are
+straightforward but tedious, and are sketched in
+Appendix~\ref{app_perturbative}. Using this method we have calculated
+$\varphi_1(\lambda)$ up to fifth order in $\tau$. In terms of the
+the dimensionless variable $x=\lambda/\overline{c}$ we have
+\bea
+\varphi_{1}(x)&=\frac{\tau^{2}}{x^2(x^2+\frac{1}{4})}
+\Bigg[1-\frac{4\tau}{x^{2}+1}+\frac{\tau^2(1+13x^2)}{(1+x^2)^2(x^2+\frac{1}{4})}-\frac{32(-1+5x^2)\tau^3}{(1+x^2)^3(1+4x^2)}\Bigg]+\mathcal{O}(\tau^{6}).
+\label{fifth_order}
+\eea
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Exact solution}
+\label{analytical}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In this section we discuss how to solve equations (\ref{coupled}),
+(\ref{coupled2}) analytically. We first observe that the distribution
+functions $\rho_n(\lambda)$ can be obtained from the set
+$\{\eta_n(\lambda)\}_n$ of functions fulfilling Eqns (\ref{coupled2}) as
+\be
+\rho_n(\lambda)=\frac{\tau}{4\pi}\frac{\partial_{\tau}\eta^{-1}_{n}(\lambda)}{1+\eta_n^{-1}(\lambda)},
+\label{rho_n}
+\ee
+where $\tau$ is given in (\ref{tau}). This relation is analogous to
+the one found in the repulsive case in Ref.~\cite{dwbc-14}. To prove
+(\ref{rho_n}) one takes the partial derivative $\partial_{\tau}$
+of both sides of (\ref{perturbative}). Combining the resulting
+equation with the thermodynamic version of the Bethe-Takahashi
+equations (\ref{coupled}), and finally invoking the uniqueness of the
+solution, we obtain (\ref{rho_n}).
+
+This leaves us with the task of solving (\ref{coupled2}). In what
+follows we introduce the dimensionless parameter
+$x=\lambda/\overline{c}$ and throughout this section, with a slight
+abuse of notation, we will use the same notations for functions of
+$\lambda$ and of $x$. Our starting point is the tri-diagonal form
+(\ref{finale_gtba}) of the coupled integral equations (\ref{coupled2}).
+Following Ref.~\cite{bwfd-14} we introduce the corresponding
+$Y$-system \cite{suzuki, kp-92}
+\be
+ y_{n}\left(x+\frac{i}{2}\right)y_{n}\left(x-\frac{i}{2}\right)=Y_{n-1}(x)Y_{n+1}(x), \qquad n\geq 1,
+\label{y-system}
+\ee
+where we define $y_0(x)=0$ and
+\be
+Y_{n}(x)=1+y_{n}(x)\,.
+\ee
+Let us now assume that there exists a set of functions
+$y_n(x)$ that satisfy the $Y$-system (\ref{y-system}), and as
+functions of the complex variable $z$ have the following properties
+\begin{enumerate}
+\item $y_{n}(z)\sim z^2$, as $z\to 0$, $\forall n\geq 1$; \label{prpty1}
+\item $y_{n}(z)$ has no poles in $-1/2<\mathrm{Im}(z)<1/2$, $\forall n\geq 1$; \label{prpty2}
+\item $y_{n}(z)$ has no zeroes in $-1/2<\mathrm{Im}(z)<1/2$ except for $z=0$, $\forall n\geq 1$. \label{prpty3}
+\end{enumerate}
+One can prove that the set of functions $y_n(x)$ with these properties
+solve the tri-diagonal form of the integral equations equations
+(\ref{finale_gtba}) \cite{bwfd-14}. To see this, one has to first take
+the logarithmic derivative of both sides of (\ref{y-system}) and take the Fourier transform, integrating in $x\in(-\infty,\infty)$. Since the argument of the
+functions in the l.h.s. is shifted by $\pm i/2$ in the imaginary
+direction, one has to use complex analysis techniques to perform the
+integral. In particular, under the assumptions (\ref{prpty1}),
+(\ref{prpty2}), (\ref{prpty3}) the application of the residue theorem
+precisely generates, after taking the inverse Fourier transform, the driving term $d(\lambda)$ in (\ref{finale_gtba}) \cite{bwfd-14}.
+
+We conjecture that the exact solution for $\eta_{1}(x)$ is given by
+\begin{equation}
+\eta_{1}(x)=\frac{x^2[1+4\tau+12\tau^2+(5+16\tau)x^2+4x^4]}{4\tau^2(1+x^2)}\,.
+\label{eta_1}
+\end{equation}
+Our evidence supporting this conjecture is as follows:
+\begin{enumerate}
+\item{} We have verified using Mathematica that the functions
+$\eta_n(x)$ generated by substituting (\ref{eta_1}) into the Y-system
+(\ref{y-system}) have the properties (\ref{prpty1}), (\ref{prpty2})
+up to $n=30$. We have checked for a substantial number of values of
+the chemical potential $h$ that they have the third property
+(\ref{prpty3}) up to $n=10$.
+\item{} Our expression (\ref{eta_1}) agrees with the expansion
+(\ref{fifth_order}) in powers of $\tau$ up to fifth order.
+\item{}
+Eqn (\ref{eta_1}) agrees perfectly with our numerical solution of the
+saddle-point equations discussed in section \ref{numerics}, as is shown
+in Fig.~\ref{distributions}.
+\end{enumerate}
+Given $\eta_1(x)$ we can use the $Y$-system (\ref{y-system}) to generate
+$\eta_{n}(x)$ with $n\geq 2$
+\bea
+\eta_{n}(x)=\frac{\eta_{n-1}\left(x+\frac{i}{2}\right)\eta_{n-1}\left(x-\frac{i}{2}\right)}{1+\eta_{n-2}(x)}-1\ , \ n\geq 2.
+\label{relation1}
+\eea
+As mentioned before, the distribution functions $\rho_{n}(x)$ can
+be obtained using (\ref{rho_n}). The explicit expressions for
+$\rho_1(x)$ and $\rho_2(x)$ are as follows:
+\bea
+ \rho_{1}(x)=\frac{2 \tau^2 (1 + x^2) (1 + 2 \tau + x^2)}{\pi (x^2 + (2 \tau + x^2)^2) (1 +
+ 5 x^2 + 4 (\tau + 3 \tau^2 + 4 \tau x^2 + x^4))},
+\eea
+\bea
+\rho_{2}(x)=\frac{16\tau^4(9+4x^2)h_1(x,\tau)}{\pi(1+4x^2+8\tau)h_2(x,\tau)h_3(x,\tau)},
+\label{eq:rho_2}
+\eea
+where
+\bea
+ h_1(x,\tau)&=&9 + 49 x^2 + 56 x^4 + 16 x^6 + 72 \tau \nonumber\\
+ &+& 168 x^2 \tau + 96 x^4 \tau + 116 \tau^2 + 176 x^2 \tau^2 + 96 \tau^3\,,\\
+ h_2(x,\tau)&=&9 + 49 x^2 + 56 x^4 + 16 x^6 + 24 \tau \nonumber \\
+ &+& 120 x^2 \tau +
+ 96 x^4 \tau + 40 \tau^2 + 160 x^2 \tau^2 + 64 \tau^3\,,\\
+ h_3(x,\tau)&=&9 x^2 + 49 x^4 + 56 x^6 + 16 x^8 + 96 x^2 \tau + 224 x^4 \tau \nonumber\\
+ &+&
+ 128 x^6 \tau + 232 x^2 \tau^2 + 352 x^4 \tau^2 +
+ 384 x^2 \tau^3 + 144 \tau^4\,.
+\eea
+The functions $\rho_n(x)$ for $n\geq 3$ are always written as rational functions but their expressions get lengthier as $n$ increases.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Physical properties of the stationary state}
+\label{phys_prop}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Local pair correlation function}
+\label{section_g2}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The distribution functions $\rho_n(\lambda)$, $\rho^h_n(\lambda)$
+completely characterize the stationary state. Their knowledge, in principle,
+allows one to calculate all local correlation functions in the thermodynamic
+limit. In practice, while formulas exist for the
+expectation values of simple local operators in the Lieb-Liniger model
+in the finite volume \cite{slavnov, ccs-07, pozsgay-11, pc-15}, it is
+generally difficult to take the thermodynamic limit of these expressions.
+In contrast to the repulsive case\cite{pozsgay-11, gs-03, kgds-03,
+ kmt-09, kci-11, nrtg-16}, much less is known in the attractive
+regime, where technical complications arise that are associated with
+the existence of string solutions to the Bethe ansatz equations. Here
+we focus on the computation of the local pair correlation function
+\be
+g_2=\frac{\langle :\hat{\rho}^2(0):\rangle}{D^2}=\frac{\langle \Psi^{\dagger}(0)\Psi^{\dagger}(0)\Psi(0)\Psi(0)\rangle}{{D^2}}.
+\label{definition_g2}
+\ee
+
+We start by applying the Hellmann-Feynman \cite{gs-03, kgds-03,
+ kci-11,mp-14} theorem to the expectation value in a general
+energy eigenstate $|\{\lambda_j\}\rangle$ with energy $E[\{\lambda_j\}]$
+of the finite system
+\be
+\langle \{\lambda_j\}| \Psi^{\dagger}\Psi^{\dagger}\Psi\Psi|\{\lambda_j\}\rangle=-\frac{1}{L}\frac{\partial E[\{\lambda_j\}]}{\partial \overline{c}}\ .
+\label{hell_fey}
+\ee
+In order to evaluate the expression on the r.h.s., we need to take the
+derivative of the Bethe-Takahashi equations (\ref{BGT})
+with respect to $\overline{c}$
+\begin{equation}
+f^{(n)}(\lambda_{\alpha})=\frac{1}{n}\sum_{m}\frac{2\pi}{L}\sum_{\beta}\left(f^{(n)}(\lambda_{\alpha})-f^{(m)}(\lambda_{\beta})-\frac{\lambda^{n}_{\alpha}}{\overline{c}}+\frac{\lambda^{m}_{\beta}}{\overline{c}}\right)a_{nm}(\lambda^{n}_{\alpha}-\lambda^{m}_{\beta})\ .
+\end{equation}
+Here $a_{nm}$ is given in Eq. (\ref{aa_function}) and
+\begin{equation}
+f^{(n)}(\lambda_{\alpha})=\frac{\partial\lambda^{n}_{\alpha}}{\partial \overline{c}}\ .
+\end{equation}
+Taking the thermodynamic limit gives
+\begin{eqnarray}
+f^{(n)}(\lambda)=\frac{2\pi}{n}\left(f^{(n)}(\lambda)-\frac{\lambda}{\overline{c}}\right)\sum_{m=1}^{\infty}\int_{-\infty}^{\infty} \mathrm{d}\mu\ \rho_{m}(\mu)a_{nm}(\lambda-\mu)&\ \nonumber \\
++ \frac{2\pi}{n}\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}\mu\ \rho_{m}(\mu)\left(\frac{\mu}{\overline{c}}-f^{(m)}(\mu)\right)a_{nm}(\lambda-\mu).&\
+\end{eqnarray}
+Using the thermodynamic version of the Bethe-Takahashi equations
+(\ref{coupled}) and defining
+\begin{equation}
+b_{n}(\lambda)=2\pi\left(\frac{\lambda}{\overline{c}}-f^{(n)}(\lambda)\right)\rho_{n}^{t}(\lambda),
+\label{a:b_function}
+\end{equation}
+we arrive at
+\begin{equation}
+b_{n}(\lambda)=n\frac{\lambda}{\overline{c}}-\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}\mu\ \frac{1}{1+\eta_{m}(\mu)}b_{m}(\mu)a_{nm}(\lambda-\mu)\ .
+\label{a:aux_1}
+\end{equation}
+The set of equations (\ref{a:aux_1}) completely fixes the functions
+$b_{n}(\lambda)$, once the functions $\eta_n(\lambda)$ are
+known. The right hand side of (\ref{hell_fey}) in the finite volume
+can be cast in the form
+\be
+\frac{\partial E}{\partial \overline{c}}=\sum_{n}\left[\sum_{\alpha}2n\lambda_{\alpha}^{n}f^{(n)}(\lambda_{\alpha})-\frac{\overline{c}}{6}n(n^2-1)\right]\ .
+\ee
+Taking the thermodynamic limit, and using (\ref{a:b_function}) we finally arrive at
+\be
+\frac{1}{L}\frac{\partial E}{\partial \overline{c}}=\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\frac{\mathrm{d}\mu}{2\pi}\ \left[2\pi \rho_{n}(\mu)\left(\frac{2n\mu^2}{\overline{c}}-\frac{\overline{c}}{6}n(n^2-1)\right)-2n\mu b_n(\mu)\frac{1}{1+\eta_{m}(\mu)}\right].
+\label{a:aux_2}
+\ee
+Combining (\ref{a:aux_1}) and (\ref{a:aux_2}) we can express the local
+pair correlation function as
+\be
+g_2(\gamma)=\gamma^2\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\ \left[2mxb_m(x)\frac{1}{1+\widetilde{\eta}_m(x)} - 2\pi \widetilde{\rho}_m(x)\left(2mx^2-\frac{m(m^2-1)}{6}\right)\right] ,
+\label{one}
+\ee
+where the functions $b_{n}(x)$ are determined by
+\be
+ b_n(x)=nx-\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}y\ \frac{1}{1+\widetilde{\eta}_{m}(y)}b_{m}(y)\widetilde{a}_{nm}(x-y).
+\label{two}
+\ee
+In (\ref{one}), (\ref{two}) we defined
+\be
+\widetilde{\eta}_{n}(x)=\eta_{n}(x\overline{c})\ ,\quad
+\widetilde{\rho}_{n}(x)=\rho_{n}(x\overline{c})\ ,\quad
+\widetilde{a}_{nm}(x)=\overline{c}a_{nm}(x\overline{c}).
+\ee
+Using the
+knowledge of the functions $\eta_n(\lambda)$ for the
+stationary state, we can solve Eqns~(\ref{two}) numerically and
+substitute the results into (\ref{one}) to obtain $g_2(\gamma)$.
+
+While (\ref{one}), (\ref{two}) cannot be solved in closed form, they
+can be used to obtain an explicit asymptotic expansion around
+$\gamma=\infty$. To that end we use (\ref{therm_momentum_energy}),
+(\ref{eq:epsilon}) and (\ref{energy}) to rewrite $g_2(\gamma)$ as
+\be
+g_2(\gamma)=2+\gamma^2\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}2mxb_m(x)\frac{1}{1+\widetilde{\eta}_m(x)}.
+\label{new_g2}
+\ee
+We then use that large values of $\gamma$ correspond to small values
+of $\tau$, cf. (\ref{fact}), and carry out a small-$\tau$ expansion of
+the functions
+\be
+\frac{1}{1+\widetilde{\eta}_n(x)}=\frac{\widetilde{\varphi}_n(x)}{(1+\widetilde{\varphi}_n(x))},
+\label{expansion_2}
+\ee
+where $\widetilde{\varphi}_n(x)=1/\widetilde{\eta}_n(x)$ as in
+section~\ref{perturbative_sec}. Substituting this expansion into the
+r.h.s. of (\ref{two}) and proceeding iteratively, we obtain an
+expansion for the functions $b_n(x)$ in powers of $\tau$. The steps
+are completely analogous to those discussed in
+section~\ref{perturbative_sec} for the functions $\varphi_n(\lambda)$
+and will not be repeated here. Finally, we use the series expansions of
+$b_n(x)$ and $(1+\widetilde{\eta}_n(x))^{-1}$ in (\ref{new_g2}) to
+obtain an asymptotic expansion for $g_2(\gamma)$. The result is
+\be
+g_{2}(\gamma)=4-\frac{40}{3\gamma}+\frac{344}{3\gamma^2}-\frac{2656}{3\gamma^3}+\frac{1447904}{225\gamma^4}+\mathcal{O}(\gamma^{-5}).
+\label{analytical_g2}
+\ee
+\begin{figure}[ht]
+\centering
+%\includegraphics[scale=0.95]{fig3.pdf}
+\includegraphics[width=\textwidth]{fig3.pdf}
+\caption{Local pair correlation function $g_2(\gamma)$ in the
+stationary state at late times after the quench. The numerical solution
+of Eqns (\ref{one}), (\ref{two}) is shown as a solid orange line. The
+asymptotic expansion (\ref{analytical_g2}) around $\gamma=\infty$
+up to order $\mathcal{O}(\gamma^{-n})$ with $n=2,3,4$ is seen to be in
+good agreement for large values of $\gamma$.}
+\label{local_pair}
+\end{figure}
+In Fig. \ref{local_pair} we compare results of a full numerical
+solution of Eqns (\ref{one}), (\ref{two}) to the asymptotic expansion
+(\ref{analytical_g2}). As expected, the latter breaks down for
+sufficiently small values of $\gamma$. In contrast to the
+large-$\gamma$ regime, the limit $\gamma\to 0$ is more difficult to
+analyze because $g_2(\gamma)$ is non-analytic in $\gamma=0$.
+The limit $\gamma\to 0$ can be calculated as shown in
+Appendix~\ref{small_gamma}, and is given by
+\be
+\lim_{\gamma\to 0}g_2(\gamma)=2.
+\label{limit_0}
+\ee
+As was already noted in Ref.~\cite{pce-16}, (\ref{limit_0}) implies
+that the function $g_2(\gamma)$ is discontinuous in
+$\gamma=0$. Indeed, $g_2(0)$ can be calculated directly by using
+Wick's theorem in the initial BEC state
+\be
+\frac{\langle {\rm BEC}|:\hat{\rho}(0)^2:|{\rm BEC}\rangle}{D^2} = 1.
+\ee
+This discontinuity, which is absent for quenches to the repulsive
+regime \cite{dwbc-14}, is ascribed to the presence of multi-particle
+bound states for all values of $\gamma\neq 0$. The former are also at
+the origin of the non-vanishing limit of $g_2(\gamma)$ for
+$\gamma\to\infty$ as it will be discussed in the next section.
+
+Finally, an interesting question is the calculation of the three-body one-point correlation function $g_3(\gamma)$ on the post-quench steady state. The latter is relevant for experimental realizations of bosons confined in one dimension, as it is proportional to the three-body recombination rate \cite{lohp-04}. For $g_3$ it is reasonable to expect that three-particle bound states may give non-vanishing contributions in the large coupling limit.
+While $g_3$ is known for general states in the repulsive Lieb-Liniger model, its computation in the attractive case is significantly harder and requires further development of existing methods. We hope that our work will motivate theoretical efforts in this direction.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Physical implications of the multi-particle bound states}
+\label{physical_discussions}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+A particularly interesting feature of our stationary state is the
+presence of finite densities of $n$-particle bound states with
+$n\geq 2$. In Fig.~\ref{strings}, their densities and energies per
+volume are shown for a number of different values of $\gamma$.
+\begin{figure}[ht]
+\centering
+%\includegraphics[scale=0.82]{fig4.pdf}
+\includegraphics[width=\textwidth]{fig4.pdf}
+\caption{ Density $D_n$ and absolute value of the normalized energies
+per volume $e_n/\gamma$ of the bosons forming $n$-particle bound
+states as defined in (\ref{density_per_string}). The plots correspond to
+$(a)$ $\gamma=20$, $(b)$ $\gamma=2$, $(c)$ $\gamma=0.2$. The total
+density is fixed $D=1$. The energy densities $e_n$ are always negative
+for $n\geq 2$ (i.e. $|e_n|=-e_n$ for $n\geq 2$) while $e_1>0$.}
+\label{strings}
+\end{figure}
+We see that the maximum of $D_n$ occurs at a value of $n$ that
+increases as $\gamma$ decreases. This result has a simple physical
+interpretation. In the attractive regime, the bosons have a tendency
+to form multi-particle bound states. One might naively expect that
+increasing the strength $\gamma$ of the attraction between bosons
+would lead to the formation of bound states with an ever increasing
+number of particles, thus leading to phase separation. However, in the
+quench setup the total energy of the system is fixed by the initial
+state, cf. (\ref{energy}), while the energy of $n$-particle
+bound states scales as $n^3$, cf. Eqns~(\ref{eq:epsilon}),
+(\ref{density_per_string}). As a result, $n$-particle bound states
+cannot be formed for large values of $\gamma$, and indeed they are
+found to have very small densities for $n\geq 3$. On the contrary,
+decreasing the interaction strength $\gamma$, the absolute value of
+their energy lowers and these bound states become accessible. The
+dependence of the peak in Fig.~\ref{strings} on $\gamma$ is monotonic
+but non-trivial and it is the result of the competition between the
+tendency of attractive bosons to cluster, and the fact that
+$n$-particle bound states with $n$ very large cannot be formed as a
+result of energy conservation.
+
+The presence of multi-particle bound states affects measurable
+properties of the system, and is the reason for the particular
+behaviour of the local pair correlation function computed in the
+previous section. Remarkably, this is true also in the limit
+$\gamma\to \infty$. This is in marked contrast to the super
+Tonks-Girardeau gas, where bound states are absent. To exhibit the
+important role of bound states in the limit of large $\gamma$, we will
+demonstrate that the limiting value of $g_2(\gamma)$ for $\gamma\to
+\infty$ is entirely determined by bound pairs. It follows from
+(\ref{one}) that $g_2(\gamma)$ can be written in the form
+\be
+g_2(\gamma)=\sum_{m=1}^{\infty}g_{2}^{(m)}(\gamma),
+\ee
+where $g_2^{(m)}(\gamma)$ denotes the contribution of
+$m$-particle bound states to the local pair correlation
+\be
+g_2^{(m)}(\gamma)= \gamma^2\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\ \left[2mxb_m(x)\frac{1}{1+\widetilde{\eta}_m(x)} - 2\pi \widetilde{\rho}_m(x)\left(2mx^2-\frac{m(m^2-1)}{6}\right)\right].
+\label{eq:temp_1}
+\ee
+Let us first show that unbound particles give a vanishing contribution
+\be
+\lim_{\gamma \to \infty}g_{2}^{(1)}(\gamma)=0.
+\label{limit1}
+\ee
+In order to prove this, we use that at leading order in $1/\gamma$ we
+have $b_1(x)=x$. Using the explicit expressions for
+$\widetilde{\eta}_1(x)$, $\widetilde{\rho}_1(x)$ we can then perform
+the integrations in the r.h.s. of Eq.~(\ref{eq:temp_1}) exactly and
+take the limit $\gamma\to \infty$ afterwards. We obtain
+\bea
+\lim_{\gamma\to\infty}\gamma^2\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\ 2xb_1(x)\frac{1}{1+\widetilde{\eta}_1(x)}=2,\\
+\lim_{\gamma\to\infty}\gamma^2\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\ \left(- 2\pi \widetilde{\rho}_1(x)2x^2\right)=-2,
+\eea
+which establishes (\ref{limit1}). Next, we address the bound pair
+contribution. At leading order in $1/\gamma$ we have $b_2(x)=2x$, and
+using the explicit expression for $\widetilde{\eta}_{2}(x)$ we obtain
+\be
+\lim_{\gamma\to\infty}\gamma^2\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\ 4xb_2(x)\frac{1}{1+\widetilde{\eta}_2(x)}=0.
+\ee
+This leaves us with the contribution
+\be
+\lim_{\gamma\to \infty} \gamma^2\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\ \left[ -2\pi \widetilde{\rho_2}(x)\left(4x^2-1\right)\right].
+\label{limit_2}
+\ee
+Although the function $\widetilde{\rho}_2(x)$ is known,
+cf. Eq.~(\ref{eq:rho_2}), its expression is unwieldy and it is
+difficult to compute the integral analytically. On the other hand, one
+cannot expand $\widetilde{\rho}_2(x)$ in $1/\gamma$ inside the
+integral, because the integral of individual terms in this expansion
+are not convergent (signalling that in this case one cannot exchange
+the order of the limit $\gamma\to\infty$ and of the
+integration). Nevertheless, the numerical computation of the integral
+in (\ref{limit_2}) for large values of $\gamma$ presents no
+difficulties and one can then compute the limit numerically. We found
+that the limit in Eq.~(\ref{limit_2}) is equal to $4$ within machine
+precision so that
+\be
+\lim_{\gamma\to\infty} g_2(\gamma)=4= \lim_{\gamma\to\infty} g_2^{(2)}(\gamma).
+\ee
+Finally, we verified that contributions coming from bound states
+with higher numbers of particles are vanishing, i.e.
+$g_2^{(m)}(\gamma)\to 0$ for $\gamma\to \infty$, $m\geq 3$. This establishes
+that the behaviour of $g_2(\gamma)$ for large values of $\gamma$ is
+dominated by bound pair of bosons.
+
+%%%%%%%%%%%%%%%%%%%%%%%
+\section{Conclusions}
+\label{conclusions}
+%%%%%%%%%%%%%%%%%%%%%%%
+We have considered quantum quenches from an ideal Bose condensate to
+the one-dimensional Lieb-Liniger model with arbitrary attractive
+interactions. We have determined the stationary state,
+and determined its physical properties. In particular, we revealed
+that the stationary state is composed of an interesting mixture of
+multi-particle bound states, and computed the local pair
+correlation function in this state. Our discussion presents a detailed
+derivation of results first announced in Ref.~\cite{pce-16}.
+
+As we have stressed repeatedly, the most intriguing feature of the
+stationary state for the quench studied in this work is the presence
+of multi-particle bound states. As was argued in Ref.~\cite{pce-16},
+their properties could in principle be probed in ultra-cold atoms
+experiments. Multi-particle bound states are also formed in the quench
+from the N\'eel state to the gapped XXZ model, as it was recently
+reported in Refs.~\cite{wdbf-14,bwfd-14,budapest,buda2}. However, in
+contrast to our case, the bound state densities are always small
+compared to the density of unbound magnons for all the values of the
+final anisotropy parameter $\Delta\geq 1$ \cite{bwfd-14}.
+
+Our work also provides an interesting physical example of a quantum
+quench, where different initial conditions lead to stationary states
+with qualitatively different features. Indeed, a quench in the
+one-dimensional Bose gas from the infinitely repulsive to the
+infinitely attractive regime leads to the super Tonks-Girardeau
+gas, where bound states are absent. On the other hand, as shown in
+section \ref{phys_prop}, if the initial state is an ideal Bose
+condensate, bound states have important consequences on the
+correlation functions of the system even in the limit of large
+negative interactions.
+
+An interesting open question is to find a description of our
+stationary state in terms of a GGE. As the stationary state involves
+bound states, it is likely that the GGE will involve not yet known
+quasi-local conserved charges \cite{idwc-15,iqdb-15,impz-16} as well as the
+known ultra-local ones\cite{davies-90}. In the Lieb-Liniger model
+technical difficulties arise when addressing such issues, as
+expectation values of local conserved charges generally exhibit divergences
+\cite{davies-90,kscc-13, kcc-14}. In addition, very little is known
+about quasi-local conserved charges for interacting models defined in
+the continuum \cite{impz-16, emp-15}.
+
+
+Finally, it would be interesting to investigate the approach to the steady state in the quench considered in this work. While this is in general a very difficult problem, in the repulsive regime the post-quench time evolution from the non-interacting BEC state was considered in \cite{dpc-15}. There an efficient numerical evaluation of the representation \eqref{time_ev} was performed, based on the knowledge of exact one-point form factors \cite{pc-15}. The attractive regime, however, is significantly more involved due to the presence of bound states and the study of the whole post-quench time evolution remains a theoretical challenge for future investigations.
+
+
+\section*{Acknowledgements}
+We thank Michael Brockmann for a careful reading of the manuscript. PC acknowledges the financial support by the ERC under Starting Grant
+279391 EDEQS. The work of FHLE was supported by the EPSRC under grant
+EP/N01930X. All authors acknowledge the hospitality of the Isaac
+Newton Institute for Mathematical Sciences under grant EP/K032208/1.
+
+
+
+% TODO: include funding information
+%\paragraph{Funding information}
+%Authors are required to provide funding information, including relevant agencies and grant numbers with linked author's initials.
+
+
+\begin{appendix}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Overlaps in the presence of zero-momentum $n$-strings}
+\label{app_overlap}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In this appendix we argue that Eq. (\ref{leading}) gives the leading
+term in the thermodynamic limit of the logarithm of the overlap
+between the BEC state and a parity-invariant Bethe state, even in
+cases where the latter contains zero-momentum strings.
+
+To see this, consider a parity invariant Bethe state
+%with a total number $\mathcal{N}_s=2K+1$ of strings,
+with a single zero-momentum $m$-string, and $K$ parity-related pairs
+of $n_j$-strings. The total number of particles in such a state is then
+$N=2\sum_{j}n_j+m$. In Ref.~\cite{cl-14} an explicit expression for
+the overlap (\ref{overlap}) of such states with a BEC state in the
+zero-density limit ($L\to \infty$ and $N$ fixed) was obtained. Up to an
+irrelevant (for our purposes) overall minus sign, it reads
+\bea
+\langle \{\lambda_{j}\}_{j=1}^{N/2}\cup \{-\lambda_{j}\}_{j=1}^{N/2} |{\rm BEC}\rangle &=& \frac{2^{m-1}L\overline{c}}{(m-1)!}\sqrt{\frac{N!}{(L\overline{c})^{N}}}\nonumber\\
+&\times&\prod_{p=1}^{K}\frac{1}{\sqrt{\frac{\lambda_p^2}{\overline{c}^2}\left(\frac{\lambda_p^2}{\overline{c}^2}+\frac{n_p^2}{4}\right)}\prod_{q=1}^{n_p-1}\left(\frac{\lambda_p^2}{\overline{c}^2}+\frac{q^2}{4}\right)},
+\label{zero_momentum_overlap}
+\eea
+where $\lambda_p$ is the centre of the $p$'th string. We see that as a
+result of having a zero-momentum string, an additional pre-factor $L$
+appears. In general, the presence of $M$ zero-momentum strings will
+lead to an additional pre-factor $L^M$ \cite{cl-14}. While
+(\ref{zero_momentum_overlap}) is derived in the zero density limit,
+we expect an additional pre-factor to be present also if one considers
+the thermodynamic limit $N,L\to \infty$, at finite density $D=N/L$.
+Importantly such pre-factors will result in \emph{sub-leading}
+corrections of order $(\ln L)/ L$ to the logarithm of the
+overlaps. This suggests that (\ref{leading}) holds even for states with
+zero-momentum $n$-strings.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Tri-diagonal form of the coupled integral equations}
+\label{app_tridiag}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Tri-diagonal Bethe-Takahashi equations}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Our starting point are the thermodynamic Bethe equations
+(\ref{coupled}). For later convenience we introduce the following
+notations for the Fourier transform of a function
+\begin{equation}
+\hat{f}(k)=\mathcal{F}[f](k)=\int_{-\infty}^{\infty}f(\lambda)e^{ik\lambda}\mathrm{d}\lambda\ ,
+\end{equation}
+\begin{equation}
+f(\lambda)=\mathcal{F}^{-1}[\hat{f}](\lambda)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{f}(k)e^{-ik\lambda}\mathrm{d}k\ .
+\end{equation}
+We recall that $f\ast g$ denotes the convolution of two functions,
+cf. (\ref{convolution}). The Fourier transform of $a_{n}(\lambda)$
+defined in (\ref{a_function}) is easily computed
+\begin{equation}
+\hat{a}_{n}(k)=e^{-\frac{n\overline{c}|k|}{2}}\ .
+\end{equation}
+Following Ref. \cite{gaudin}, we introduce the symbols
+\begin{equation}
+[nmp]=\left\{\begin{array}{cc}1\ , & \mathrm{if}\ p=|m-n|\ \mathrm{or}\ m+n\\2\ , & \mathrm{if}\ p= |m-n|+2,\ |m-n|+4, \ldots, m+n-2\ , \\0 & \mathrm{otherwise}\ .\end{array}\right.
+\end{equation}
+We can then perform the Fourier transform of both sides of (\ref{coupled}) and obtain
+\begin{equation}
+n\delta(k)-\sum_{m=1}\sum_{p>0}[nmp]\hat{\rho}_{m}(k)e^{-\frac{\overline{c}}{2}|k|p}=\hat{\rho}_n^t(k)\ ,
+\label{a:intermediate}
+\end{equation}
+where $\rho_n^t(\lambda)$ are given in (\ref{rho_tot}). We now define
+\begin{eqnarray}
+\hat{\rho}_{-m}(k)=-\hat{\rho}_{m}(k)\ ,\qquad m\geq 1 ,&\\
+\hat{\rho}_{0}(k)=0\ . &
+\end{eqnarray}
+After straightforward calculations, we can rewrite (\ref{a:intermediate}) in the form
+\begin{equation}
+\hat{\rho}_{n}^h(k)=n\delta(k)-\coth\left(\frac{|k|\overline{c}}{2}\right)\sum_{m=-\infty}^{+\infty}e^{-|k||n-m|\frac{\overline{c}}{2}}\hat{\rho}_{m}(k)\ .
+\label{temp_1}
+\end{equation}
+In order to decouple these equations we note that
+\begin{eqnarray}
+\hat{\rho}_{n+1}^{h}(k)&+&\hat{\rho}_{n-1}^{h}(k)=2n\delta(k)\nonumber \\
+&-&\coth\left(\frac{|k|\overline{c}}{2}\right)\left[-2\hat{\rho}_{n}(k)\sinh\left(\frac{|k|\overline{c}}{2}\right)+2\cosh\left(\frac{|k|\overline{c}}{2}\right)\sum_{m=-\infty}^{\infty}e^{-|k||n-m|\frac{\overline{c}}{2}}\hat{\rho}_{m}(k)\right]\ .\nonumber \\ \label{temp_2}
+\end{eqnarray}
+Combining Eqns (\ref{temp_1}), (\ref{temp_2}) one obtains
+\begin{eqnarray}
+\hat{\rho}_{n}^t(k)&=&\frac{1}{2\cosh\left(|k|\overline{c}/2\right)}\left(\hat{\rho}_{n+1}^h(k)+\hat{\rho}_{n-1}^h(k)\right)-\underbrace{n\delta(k)\left[\frac{1-\cosh\left(\frac{|k|\overline{c}}{2}\right)}{\cosh\left(\frac{|k|\overline{c}}{2}\right)}\right]}_{=0}=\nonumber\\
+&=&\frac{1}{2\cosh\left(|k|\overline{c}/2\right)}\left(\hat{\rho}_{n+1}^h(k)+\hat{\rho}_{n-1}^h(k)\right)\ .
+\end{eqnarray}
+We can now perform the inverse Fourier transform. Using
+\begin{equation}
+\frac{1}{2\pi}\int_{-\infty}^{\infty}\ d k\frac{1}{\cosh\left(k\frac{\overline{c}}{2}\right)}e^{-i\lambda k}=\frac{1}{\overline{c}}\frac{1}{\cosh\left(\frac{\lambda\pi}{\overline{c}}\right)}\ ,
+\label{a:integral}
+\end{equation}
+we finally obtain
+\begin{eqnarray}
+ \rho_{n}(1+\eta_{n})=s\ast\left(\eta_{n-1}\rho_{n-1}+\eta_{n+1}\rho_{n+1}\right)\qquad n\geq 1\ , \label{a:final_bethe}
+\label{a:final_bethe_2}
+\end{eqnarray}
+where we can choose $\eta_{0}(\lambda)\rho_0(\lambda)=\delta(\lambda)$, $\eta_n(\lambda)$ is given in Eq.~(\ref{eq:eta}), and where
+\begin{equation}
+s(\lambda)=\frac{1}{2\overline{c}\cosh\left(\frac{\pi\lambda}{\overline{c}}\right)}\ .
+\label{a:kernel}
+\end{equation}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Tri-diagonal oTBA equations}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In this appendix we derive the tri-diagonal equations
+(\ref{finale_gtba}) starting from Eqns (\ref{coupled2}). Our
+discussion follows Ref.~\cite{wdbf-14}. Some useful identities are\cite{takahashi}
+\begin{equation}
+(a_0+a_2)\ast a_{nm}=a_1\ast(a_{n-1,m}+a_{n+1,m})+(\delta_{n-1,m}+\delta_{n+1,m})a_1\ ,\qquad n\geq 2,\ m\geq 1,
+\end{equation}
+\begin{equation}
+(a_0+a_2)\ast a_{1m}=a_1\ast a_{2,m}+a_1\delta_{2,m}\ , \qquad m\geq 1\ ,
+\end{equation}
+where we define $a_0(\lambda)=\delta(\lambda)$, and where the
+functions $a_{nm}(\lambda)$, $a_n(\lambda)$ are given in
+Eqns~(\ref{aa_function}), (\ref{a_function}). Convolution of
+(\ref{coupled2}) with $(a_0+a_2)$ gives
+\begin{eqnarray}
+(a_0+a_2)\ast\ln\eta_n&=&(a_0+a_2)\ast g_n-a_1\ast (g_{n-1}+g_{n+1})\nonumber \\
+&+&a_{1}\ast\left[\ln(1+\eta_{n-1})+\ln(1+\eta_{n+1})\right]\ ,\qquad n\geq 1\ ,
+\label{a:simplified}
+\end{eqnarray}
+where we defined $g_{n}(\lambda)=-\ln W_n(\lambda)$, $g_0(\lambda)=0$ and
+$\eta_0(\lambda)=0$. The functions $g_n(\lambda)$ can be written as
+\begin{equation}
+g_{n}(\lambda)=\ln s_{0}^{(2)}(\lambda)+\ln s_{n}^{(2)}(\lambda)+2\sum_{\ell=1}^{n-1}\ln s_{\ell}^{(2)}(\lambda)\ ,
+\label{a:gn}
+\end{equation}
+where
+\begin{equation}
+s_{\ell}^{(2)}(\lambda)=s_{\ell}(\lambda)s_{-\ell}(\lambda)=\frac{\lambda^2}{\overline{c}^2}+\frac{\ell^2}{4}\ .
+\end{equation}
+%
+It is straightforward to show that
+\begin{equation}
+(a_{m}\ast f_{r})(\lambda)=f_{m+r}(\lambda)\,,
+\label{a:identity}
+\end{equation}
+where we defined
+\be
+f_{r}(\lambda)=\ln\left[\left(\frac{\lambda}{\overline{c}}\right)^2+\left(\frac{r}{2}\right)^2\right]\,.
+\label{ffunction}
+\ee
+Using (\ref{a:identity}) and (\ref{a:gn}), we can rewrite the driving
+term in (\ref{a:simplified}) as
+\begin{eqnarray}
+\tilde{d}_n\equiv (a_0+a_{2})\ast g_n-a_1\ast (g_{n-1}+g_{n+1})=f_{0}-f_{2}=\ln\left(\frac{\lambda^2}{\overline{c}^2}\right)-\ln\left(\frac{\lambda^2}{\overline{c}^2}+1\right)\ ,&
+\label{a:important1}
+\end{eqnarray}
+which allows us to rewrite the oTBA equations in the form
+\begin{equation}
+(a_0+a_2)\ast \ln\eta_n=\tilde{d}_n+a_{1}\ast\left[\ln(1+\eta_{n-1})+\ln(1+\eta_{n+1})\right]\ .
+\label{a:important2}
+\end{equation}
+We note that $\tilde{d}_n$ is in fact independent of $n$. Carrying out the
+Fourier transform and using that $f_0-f_2=(a_0-a_2)\ast f_0$ we obtain
+\begin{eqnarray}
+\mathcal{F}\left[\ln \eta_n\right]&=&\frac{1}{1+e^{-\overline{c}|k|}}(1-e^{-\overline{c}|k|})\mathcal{F}\left[f_0\right]\nonumber \\
+&+&\frac{1}{1+e^{-\overline{c}|k|}}e^{-\frac{\overline{c}|k|}{2}}\mathcal{F}\left[\left(\ln(1+\eta_{n-1})+\ln(1+\eta_{n+1})\right)\right]\ .
+\label{a:aa}
+\end{eqnarray}
+The first term on the right hand side simplifies
+\begin{equation}
+\frac{1}{1+e^{-\overline{c}|k|}}(1-e^{-\overline{c}|k|})\mathcal{F}\left[f_0\right]=-2\pi\frac{\tanh(\overline{c}k/2)}{k}\ .
+\end{equation}
+Finally, taking the inverse Fourier transform of (\ref{a:aa}), using
+(\ref{a:integral}) as well as
+\begin{equation}
+\int_{-\infty}^{\infty} d k e^{-i k \lambda} \frac{\tanh(\overline{c}k/2)}{k}=-\ln\left[\tanh^2\left(\frac{\pi\lambda}{2\overline{c}}\right)\right]\ ,
+\end{equation}
+we arrive at the desired tri-diagonal form of the oTBA equations
+\begin{eqnarray}
+ \ln(\eta_n)=d+s\ast\left[\ln(1+\eta_{n-1})+\ln(1+\eta_{n+1})\right]\ ,\qquad n\geq 1\ , &\\
+ \eta_{0}(\lambda)=0\ .&
+\label{a:finale gtba}
+\end{eqnarray}
+Here $s(\lambda)$ is given by Eq. (\ref{a:kernel}) and
+\be
+d(\lambda)=\ln\left[\tanh^2\left(\frac{\pi\lambda}{2\overline{c}}\right)\right]. \ee
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Asymptotic behaviour}
+\label{app_asymptotic}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In this appendix we derive the asymptotic condition (\ref{difference})
+for the tri-diagonal equations (\ref{finale_gtba}). Our derivation closely
+follows the finite temperature case \cite{takahashi}. We start from
+Eq.~(\ref{coupled2}) for $n=1$
+\be
+\ln \eta_{1}(\lambda)=-2h+(f_0+f_1)+a_2\ast\ln(1+\eta^{-1}_{1})+\sum_{m=2}^{+\infty}(a_{m-1}+a_{m+1})\ast\ln\left(1+\eta^{-1}_{m}\right)\ ,
+\label{case1}
+\ee
+where $f_{r}=f_{r}(\lambda)$ is defined in (\ref{ffunction}). We use now the following identities, which are easily derived from (\ref{a:identity}), (\ref{a:important1}), (\ref{a:important2})
+\begin{eqnarray}
+a_2\ast \ln (1+\eta_{1}^{-1})&=&a_2\ast \ln(1+\eta_1)-a_2\ast \ln \eta_1=\nonumber \\
+&=&a_2\ast \ln(1+\eta_1)-f_0+f_2-a_1\ast \ln (1+\eta_2)+\ln \eta_1\ .
+\label{passaggio}
+\end{eqnarray}
+Using (\ref{passaggio}) we can recast (\ref{case1}) in the form
+\begin{eqnarray}
+-2h+a_1\ast(f_0+f_1)=a_1\ast \ln \eta_2&-&a_2\ast\ln(1+\eta_1)-a_3\ast\ln(1+\eta_2^{-1})\nonumber \\
+ &-&\sum_{m=3}^{+\infty}(a_{m-1}+a_{m+1})\ast\ln(1+\eta_m^{-1})\ .
+\label{eq:aaa}
+\end{eqnarray}
+To proceed, we write
+\bea
+\sum_{m=3}^{+\infty}(a_{m-1}+a_{m+1})\ast\ln(1+\eta_m^{-1})
+&=&(a_{2}+a_{4})\ast\ln(1+\eta_{3}^{-1}) \nonumber \\ &+&\sum_{m=4}^{+\infty}(a_{m-1}+a_{m+1})\ast\ln(1+\eta_m^{-1})\ .
+\eea
+After rewriting the first term on the right hand side, we substitute
+back into (\ref{eq:aaa}) to obtain
+\bea
+-2h+a_2\ast(f_0+f_1)=a_2\ast \ln \eta_3&-&a_3\ast\ln(1+\eta_2)-a_4\ast\ln(1+\eta_3^{-1})\nonumber \\
+&-&\sum_{m=4}^{+\infty}(a_{m-1}+a_{m+1})\ast\ln(1+\eta_m^{-1}).
+\label{eq:aab}
+\eea
+Iterating the above procedure $n$ times we arrive at
+\bea
+-2h+a_n\ast(f_0+f_1)=a_n\ast \ln \eta_{n+1}&-&a_{n+1}\ast\ln(1+\eta_{n})-a_{n+2}\ast\ln(1+\eta_{n+1}^{-1})\nonumber \\
+&-&\sum_{m=n+2}^{+\infty}(a_{m-1}+a_{m+1})\ast\ln(1+\eta_m^{-1})\ .
+\label{eq:aac}
+\eea
+Fourier transforming and using the definition for $f_r$ given in
+(\ref{ffunction}) we obtain
+\bea
+ \ln\eta_{n+1}&=&-2h+\ln\left[\frac{\lambda}{\overline{c}}\left(\frac{\lambda^2}{\overline{c}^2}+\frac{1}{4}\right)\right]+a_1\ast\ln\eta_n\nonumber \\
+ &&\hspace{-10mm} + a_{1}\ast\ln(1+\eta_{n}^{-1})+a_{2}\ast\ln(1+\eta_{n+1}^{-1})+\sum_{m=2}^{+\infty}(a_{m-1}+a_{m+1})\ast\ln(1+\eta_{m+n}^{-1}).
+\label{almost_final}
+\eea
+Assuming that $\eta_{n}^{-1}(\lambda)$ is vanishing sufficiently fast
+as $n\to \infty$ for a generic (and fixed) value of $\lambda$, we can
+drop the infinite sum and the two previous terms, and arrive at
+Eq.~(\ref{difference}).
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Perturbative analysis}
+\label{app_perturbative}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In this appendix we sketch the calculations leading to the expansion (\ref{fifth_order}). Throughout this appendix we work with the dimensionless variable $x=\lambda/\overline{c}$. At the lowest order, it follows from Eq.~(\ref{coefficient}) that
+\be
+\varphi_1(x)=\frac{\tau^2}{x^2\left(x^2+\frac{1}{4}\right)}+\mathcal{O}(\tau^3).
+\label{second_order}
+\ee
+Since $\varphi_n(x)\propto \tau^{2n}$, we can neglect $\varphi_n(x)$
+with $n\geq 2$ to compute the third order expansion of
+$\varphi_1(x)$. Hence, the infinite sum in (\ref{perturbative}) for
+$n=1$ can be truncated, at third order in $\tau$, to the first term
+($m=1$), where we can use the expansion (\ref{second_order}) for
+$\varphi_{1}(\lambda)$. Following Ref.~\cite{dwbc-14} one can then use
+identity (\ref{a:identity}) to perform the convolution integral and
+finally obtain
+\be
+\varphi_1(x)=\frac{\tau^2}{x^2\left(x^2+\frac{1}{4}\right)}\left(1-\frac{4\tau}{x^2+1}\right)+\mathcal{O}(\tau^4).
+\ee
+One can then perform the same steps for higher order corrections, at
+each stage of the calculation keeping all the relevant terms. For
+example, already at the fourth order in $\tau$ of $\varphi_1(x)$ one
+cannot neglect the lowest order contribution coming from
+$\varphi_2(x)$ in the r.h.s. of Eq.~(\ref{perturbative}). For higher
+orders one also has to consider corrections to $\varphi_n(x)$ with
+$n\geq 2$.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Small $\gamma$ limit for $g_2$}
+\label{small_gamma}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In this appendix we prove that
+\begin{equation}
+\lim_{\gamma\to 0}g_2(\gamma)=2\ .
+\label{a:to_prove}
+\end{equation}
+Our starting point is Eqn (\ref{new_g2}). Rescaling variables by
+\begin{eqnarray}
+\hat{b}_{m}(x)=\sqrt{\gamma}b_{m}\left(\frac{x}{\sqrt{\gamma}}\right)\ ,\qquad \hat{\eta}_{n}(x)=\widetilde{\eta}_{n}\left(\frac{x}{\sqrt{\gamma}}\right)\ ,
+\end{eqnarray}
+we have
+\begin{eqnarray}
+g_2=2+ \sqrt{\gamma}\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\ \left[2mx\hat{b}_m(x)\frac{1}{1+\hat{\eta}_m(x)}\right]\,.
+\label{a:asympt}
+\end{eqnarray}
+The functions $\hat{b}_{n}(x)$ satisfy the coupled nonlinear integral equations
+\begin{equation}
+\hat{b}_n(x)=nx-\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}y\ \frac{1}{1+\hat{\eta}_{m}(y)}\hat{b}_{m}(y)\hat{a}_{nm}(x-y)\ ,
+\end{equation}
+where
+\begin{equation}
+\hat{a}_{nm}(x)=\frac{1}{\sqrt{\gamma}}\widetilde{a}_{nm}\left(\frac{x}{\gamma}\right).
+\end{equation}
+Our goal is to determine the limit
+\begin{equation}
+\lim_{\gamma\to 0}\sum_{m=1}^{\infty}\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{2\pi}\ \left[2mx\hat{b}_m(x)\frac{1}{1+\hat{\eta}_m(x)}\right]\ .
+\end{equation}
+The calculation is non-trivial as we cannot exchange the infinite sum
+with the limit. However, based on numerical evidence we claim that
+this limit is finite, and (\ref{a:to_prove}) then immediately follows
+from (\ref{a:asympt}).
+
+Note that the numerical computation of $g_2(\gamma)$ is increasingly
+demanding as $\gamma\to 0$, due to the fact that more and more strings
+contribute. Accordingly, the infinite systems (\ref{one}) and
+(\ref{two}) have to be truncated to a larger number of equations and
+the numerical computation takes more time to provide precise results.
+We were able to numerically compute $g_2(\gamma)$ for decreasing
+values of $\gamma$ down to $\gamma=0.025$ where $g_2(0.025)\simeq
+2.11$ and approximately $30$ strings contributed to the
+computation. We fitted the numerical data for small $\gamma$ with
+$G(\gamma)=\alpha_1+\alpha_2\sqrt{\gamma}$ and we correctly found
+$\alpha_1=2$ within the numerical error.
+
+\end{appendix}
+
+% TODO:
+% Provide your bibliography here, either by:
+% - writing it directly, following the example below, including Author(s), Title, Journal Ref. with year in parentheses at the end, followed by the DOI number.
+% - running BiBTeX using the class SciPost_bibstyle_v1.bst and pasting the .bbl file contents here.
+\begin{thebibliography}{999}
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+
+
+%------ Quench action-------
+
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+
+%-----super-Tonks-Girardeau--------------
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