Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2007.12683

Non-Equilibrium Steady State of the Lieb-Liniger model: exact treatment of the Tonks Girardeau limit

Spyros Sotiriadis

Published Jul 24, 2020• 2 citations
arXivHTMLPDF

Abstract

Aiming at studying the emergence of Non-Equilibrium Steady States (NESS) in quantum integrable models by means of an exact analytical method, we focus on the Tonks-Girardeau or hard-core boson limit of the Lieb-Liniger model. We consider the abrupt expansion of a gas from one half to the entire confining box, a prototypical case of inhomogeneous quench, also known as "geometric quench". Based on the exact calculation of quench overlaps, we develop an analytical method for the derivation of the NESS by rigorously treating the thermodynamic and large time and distance limit. Our method is based on complex analysis tools for the derivation of the asymptotics of the many-body wavefunction, does not make essential use of the effectively non-interacting character of the hard-core boson gas and is sufficiently robust for generalisation to the genuinely interacting case.

DOI
10.48550/arxiv.2007.12683

Related papers

Non-Equilibrium Steady State of the Lieb-Liniger model: multiple-integral representation of the time evolved many-body wave-function

Spyros Sotiriadis

We continue our study of the emergence of Non-Equilibrium Steady States in quantum integrable models focusing on the expansion of a Lieb-Liniger gas for arbitrary repulsive interaction. As a first step towards the derivation of the asymptotics of observables in the thermodynamic and large distance and time limit, we derive an exact multiple integral representation of the time evolved many-body wave-function. Starting from the known but complicated expression for the overlaps of the initial state of a geometric quench, which are derived from the Slavnov formula for scalar products of Bethe states, we eliminate the awkward dependence on the system size and distinguish the Bethe states into convenient sectors. These steps allow us to express the rather impractical sum over Bethe states as a multiple rapidity integral in various alternative forms. Moreover, we examine the singularities of the obtained integrand and calculate the contribution of the multivariable kinematical poles, which is essential information for the derivation of the asymptotics of interest.

arXiv:2010.03553•Oct 7, 2020•5 citations

Relaxation dynamics of the Lieb-Liniger gas following an interaction quench: A coordinate Bethe-ansatz analysis

Jan C. Zill, Tod M. Wright, Karen V. Kheruntsyan +2 more

We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.

arXiv:1407.4998•Physical Review A•Feb 12, 2015•43 citations

Quenches from bosonic Gaussian initial states to the Tonks-Girardeau limit: Stationary states and effects of a confining potential

Alvise Bastianello, Mario Collura, Spyros Sotiriadis

We consider the general problem of quenching an interacting Bose gas from the noninteracting regime to the strongly repulsive limit described by the Tonks-Girardeau gas with the initial state being a Gaussian ensemble in terms of the bosons. A generic multipoint correlation function in the steady state can be described fully in terms of a Fredholm-like determinant suitable both for a numerical and for an analytical study in certain limiting cases. Finally, we extend the study to the presence of a smooth confining potential showing that, in the thermodynamic limit, the time evolution of the two-point function can be mapped to a classical problem.

arXiv:1702.01729•Physical review. B./Physical review. B•May 23, 2017•19 citations

On the Lieb-Liniger model in the infinite coupling constant limit

Stéphane Ouvry, Alexios P. Polychronakos

We consider the one-dimensional Lieb-Liniger model (bosons interacting via 2-body delta potentials) in the infinite coupling constant limit (the so-called Tonks-Girardeau model). This model might be relevant as a description of atomic Bose gases confined in a one-dimensional geometry. It is known to have a fermionic spectrum since the N-body wavefunctions have to vanish at coinciding points, and therefore be symmetrizations of fermionic Slater wavefunctions. We argue that in the infinite coupling constant limit the model is indistinguishable from free fermions, i.e., all physically accessible observables are the same as those of free fermions. Therefore, Bose-Einstein condensate experiments at finite energy that preserve the one-dimensional geometry cannot test any bosonic characteristic of such a model.

•Jan 1, 2009•5 citations

Exact dynamics following an interaction quench in a one-dimensional anyonic gas

Lorenzo Piroli, Pasquale Calabrese

We study the nonequilibrium quench dynamics of a one-dimensional anyonic gas. We focus on the integrable anyonic Lieb-Liniger model and consider the quench from noninteracting to hard-core anyons. We study the dynamics of the local properties of the system. By means of integrability-based methods, we compute analytically the one-body density matrix and the density-density correlation function at all times after the quench and in particular at infinite time. Our results show that the system evolves from an initial state where the local momentum distribution function is nonsymmetric to a steady state where it becomes symmetric. Furthermore, while the initial momentum distribution functions (and the equilibrium ones) explicitly depend on the anyonic parameter, the final ones do not. This is reminiscent of the dynamical fermionization observed in the context of free expansions after release from a confining trap.

arXiv:1705.06470•Physical Review A•Aug 9, 2017•51 citations

Failure of the Quench Action Formalism for Mott Insulator Initial States

Garry Goldstein

The quench action formalism relies on the assumption that the overlap between a generic initial state $\left|Ψ_{0}\right\rangle $ and an eigenstate of an integrable model - defined through the rapidities $\left|k_{1},...k_{N}\right\rangle $ - can be written as: \begin{equation} \left\langle k_{1},...k_{N}\midΨ_{0}\right\rangle =\exp\left(-S_{Ψ_{0}}\left(ρ\left(k\right)\right)\right),\label{eq:Exponential} \end{equation} where $ρ\left(k\right)$ is the quasiparticle density of the state $\left|k_{1},...k_{N}\right\rangle $ and $S_{Ψ_{0}}$ is some smooth function of $ρ\left(k\right)$ that depends on $Ψ_{0}$. In particular the quench action formalism assumes the overlap depends smoothly on the quasiparticle density $ρ\left(k\right)$. In this work, by explicit counter example, we show that this is not the case. We consider the quench between a Mott insulator and a Lieb Liniger gas. We show that the overlap between the ground state of the Mott insulator and arbitrary eigenstates of the Lieb Liniger gas has a highly singular behavior and no expression like Eq. (1) applies. We do so within the Tonks Girardeau limit of the Lieb Liniger gas and to leading order in the $1/c$ expansion for the overlap (with $c$ being the coupling constant of the Lieb Liniger gas). In the Appendix we show similar results for overlaps in the XXZ model with crystal states.

arXiv:2606.07111•Jun 5, 2026•0 citations
Aiming at studying the emergence of Non-Equilibrium Steady States (NESS) in quantum integrable models by means of an exact analytical method, we focus on the Tonks-Girardeau or hard-core boson limit of the Lieb-Liniger model. We consider the abrupt expansion of a gas from one half to the entire confining box, a prototypical case of inhomogeneous quench, also known as "geometric quench". Based on the exact calculation of quench overlaps, we develop an analytical method for the derivation of the NESS by rigorously treating the thermodynamic and large time and distance limit. Our method is based on complex analysis tools for the derivation of the asymptotics of the many-body wavefunction, does not make essential use of the effectively non-interacting character of the hard-core boson gas and is sufficiently robust for generalisation to the genuinely interacting case.
We continue our study of the emergence of Non-Equilibrium Steady States in quantum integrable models focusing on the expansion of a Lieb-Liniger gas for arbitrary repulsive interaction. As a first step towards the derivation of the asymptotics of observables in the thermodynamic and large distance and time limit, we derive an exact multiple integral representation of the time evolved many-body wave-function. Starting from the known but complicated expression for the overlaps of the initial state of a geometric quench, which are derived from the Slavnov formula for scalar products of Bethe states, we eliminate the awkward dependence on the system size and distinguish the Bethe states into convenient sectors. These steps allow us to express the rather impractical sum over Bethe states as a multiple rapidity integral in various alternative forms. Moreover, we examine the singularities of the obtained integrand and calculate the contribution of the multivariable kinematical poles, which is essential information for the derivation of the asymptotics of interest.
We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.
We consider the general problem of quenching an interacting Bose gas from the noninteracting regime to the strongly repulsive limit described by the Tonks-Girardeau gas with the initial state being a Gaussian ensemble in terms of the bosons. A generic multipoint correlation function in the steady state can be described fully in terms of a Fredholm-like determinant suitable both for a numerical and for an analytical study in certain limiting cases. Finally, we extend the study to the presence of a smooth confining potential showing that, in the thermodynamic limit, the time evolution of the two-point function can be mapped to a classical problem.
We consider the one-dimensional Lieb-Liniger model (bosons interacting via 2-body delta potentials) in the infinite coupling constant limit (the so-called Tonks-Girardeau model). This model might be relevant as a description of atomic Bose gases confined in a one-dimensional geometry. It is known to have a fermionic spectrum since the N-body wavefunctions have to vanish at coinciding points, and therefore be symmetrizations of fermionic Slater wavefunctions. We argue that in the infinite coupling constant limit the model is indistinguishable from free fermions, i.e., all physically accessible observables are the same as those of free fermions. Therefore, Bose-Einstein condensate experiments at finite energy that preserve the one-dimensional geometry cannot test any bosonic characteristic of such a model.
We study the nonequilibrium quench dynamics of a one-dimensional anyonic gas. We focus on the integrable anyonic Lieb-Liniger model and consider the quench from noninteracting to hard-core anyons. We study the dynamics of the local properties of the system. By means of integrability-based methods, we compute analytically the one-body density matrix and the density-density correlation function at all times after the quench and in particular at infinite time. Our results show that the system evolves from an initial state where the local momentum distribution function is nonsymmetric to a steady state where it becomes symmetric. Furthermore, while the initial momentum distribution functions (and the equilibrium ones) explicitly depend on the anyonic parameter, the final ones do not. This is reminiscent of the dynamical fermionization observed in the context of free expansions after release from a confining trap.
The quench action formalism relies on the assumption that the overlap between a generic initial state $\left|Ψ_{0}\right\rangle $ and an eigenstate of an integrable model - defined through the rapidities $\left|k_{1},...k_{N}\right\rangle $ - can be written as: \begin{equation} \left\langle k_{1},...k_{N}\midΨ_{0}\right\rangle =\exp\left(-S_{Ψ_{0}}\left(ρ\left(k\right)\right)\right),\label{eq:Exponential} \end{equation} where ρ(k)ρ\left(k\right)ρ(k) is the quasiparticle density of the state $\left|k_{1},...k_{N}\right\rangle $ and SΨ0S_{Ψ_{0}}SΨ0​​ is some smooth function of ρ(k)ρ\left(k\right)ρ(k) that depends on Ψ0Ψ_{0}Ψ0​. In particular the quench action formalism assumes the overlap depends smoothly on the quasiparticle density ρ(k)ρ\left(k\right)ρ(k). In this work, by explicit counter example, we show that this is not the case. We consider the quench between a Mott insulator and a Lieb Liniger gas. We show that the overlap between the ground state of the Mott insulator and arbitrary eigenstates of the Lieb Liniger gas has a highly singular behavior and no expression like Eq. (1) applies. We do so within the Tonks Girardeau limit of the Lieb Liniger gas and to leading order in the 1/c1/c1/c expansion for the overlap (with ccc being the coupling constant of the Lieb Liniger gas). In the Appendix we show similar results for overlaps in the XXZ model with crystal states.