From 96935ca099c7e7fb2caa585e07c03540aa999c7b Mon Sep 17 00:00:00 2001 From: "J.-S. Caux" <J.S.Caux@uva.nl> Date: Tue, 12 Oct 2021 09:50:37 +0200 Subject: [PATCH] Add source files and checksum --- .gitignore | 14 + CROSSMARK_BW_square_no_text.png | Bin 0 -> 22941 bytes LICENSE | 395 ++++++ README.md | 28 +- SciPost.cls | 87 ++ SciPostPhys_1_1_001.pdf | Bin 0 -> 463234 bytes SciPostPhys_1_1_001.tex | 2019 +++++++++++++++++++++++++++++++ by.eps | 280 +++++ fig1.pdf | Bin 0 -> 123478 bytes fig3.pdf | Bin 0 -> 36922 bytes fig4.pdf | Bin 0 -> 37918 bytes logo_scipost_with_bgd.pdf | Bin 0 -> 41874 bytes sha256_checksum.txt | 11 + 13 files changed, 2832 insertions(+), 2 deletions(-) create mode 100644 .gitignore create mode 100644 CROSSMARK_BW_square_no_text.png create mode 100644 LICENSE create mode 100644 SciPost.cls create mode 100644 SciPostPhys_1_1_001.pdf create mode 100644 SciPostPhys_1_1_001.tex create mode 100644 by.eps create mode 100644 fig1.pdf create mode 100644 fig3.pdf create mode 100644 fig4.pdf create mode 100644 logo_scipost_with_bgd.pdf create mode 100644 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For +the avoidance of doubt, this paragraph does not form part of the public +licenses. + +Creative Commons may be contacted at creativecommons.org. diff --git a/README.md b/README.md index df5b286..7aeecb0 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 0000000..b4b1132 --- /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 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+% 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 + } +} + +\urlstyle{sf} + +\fancypagestyle{SPtitlepage}{% +\fancyhf{} +\fancyfoot[C]{\textbf{\thepage}} +\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)}} +\renewcommand{\headrulewidth}{1pt} +} + +\fancypagestyle{SPbulk}{ +\fancyhf{} +\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. 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