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

Physical Review A

One-body dynamical correlation function of the Lieb-Liniger model at finite temperature

Song Cheng, Yang-Yang Chen, Xi-Wen Guan, Wen‐Li Yang, Hai-Qing Lin

Published Jan 17, 2025• 4 citations
Publisher

Abstract

Building on quantum integrability, the authors address the finite temperature one-body dynamical correlation function of 1D Bose gases using an algorithm grounded in the concept of generalized particle-hole excitations relative to a reference state. The correlation function is analyzed from the momentum-energy plane and its line-shape and momentum distribution are presented, well beyond the applicable regime of Tomonaga-Luttinger liquid theory predictions.

DOI
10.1103/physreva.111.l010802

Related papers

One-body dynamical correlation function of Lieb-Liniger model at finite temperature

Song Cheng, Yang-Yang Chen, Xi‐Wen Guan +2 more

The dynamical correlated properties of one-dimensional (1D) Bose gases provide profound understanding of novel physics emergent from collective excitations, for instance, the breakdown of off-diagonal long-range order, and the establishment of Tomonaga-Luttinger liquid theory. However, due to the nonperturbative nature of 1D many-body systems, the exact evaluation of correlation functions is notoriously difficult. Here, by means of a form factor approach based on an algebraic Bethe ansatz and numerics, we present a thorough study on the one-body dynamical correlation function (1BDCF) of the Lieb-Liniger model at finite temperature. The influence of thermal fluctuation and interaction on the behavior of 1BDCF has been demonstrated and analyzed from various perspectives, including the spectral distribution, the line shape at fixed momentum, and the corresponding static correlations.

arXiv:2211.00282•Nov 1, 2022•3 citations

Universal local pair correlations of Lieb-Liniger bosons at quantum criticality

Ming‐Ming Wang, Jiahao Huang, Chaohong Lee +3 more

The one-dimensional Lieb-Liniger Bose gas is a prototypical many-body system featuring universal Tomonaga-Luttinger liquid (TLL) physics and free fermion quantum criticality. We analytically calculate finite temperature local pair correlations for the strong-coupling Bose gas at quantum criticality using the polylog function in the framework of the Yang-Yang thermodynamic equations. We show that the local pair correlation has the universal value ${g}^{(2)}(0)\ensuremath{\approx}2p/(n\ensuremath{\varepsilon})$ in the quantum critical regime, the TLL phase, and the quasiclassical region, where $p$ is the pressure per unit length rescaled by the interaction energy $\ensuremath{\varepsilon}=\frac{{\ensuremath{\hbar}}^{2}}{2m}{c}^{2}$ with interaction strength $c$ and linear density $n$. This suggests the possibility to test finite temperature local pair correlations for the TLL in the relativistic dispersion regime and to probe quantum criticality with the local correlations beyond the TLL phase. Furthermore, thermodynamic properties at high temperatures are obtained by both high temperature and virial expansion of the Yang-Yang thermodynamic equation.

arXiv:1211.3267•Physical Review A•Apr 29, 2013•14 citations

Exact spectral function of one-dimensional Bose gases

Song Cheng, Yang-Yang Chen, Xi-Wen Guan +3 more

Exactly solved models provide rigorous understanding of many-body phenomena in strongly correlated systems. In this article, we report a breakthrough in uncovering universal many-body correlated properties of the quantum integrable Lieb-Liniger model. We exactly calculate the dynamical correlation functions by computing the form factors through a newly developed method, by which we are capable of calculating all possible 'relative excitations' over the ground state or a finite temperature state to high precision. Consequently, full spectral functions obtained for the model manifest the unique power-law singularity behaviour at the spectral threshold, confirming the validity of nonlinear Luttinger liquid theory. Our method advances the theory of dynamical correlation functions with high precision towards the thermodynamic limit, and is capable of benchmarking experimental observation of such novel correlated properties.

arXiv:2209.15221•National Science Review•Jul 21, 2025•4 citations

Split Fermi seas in one-dimensional Bose fluids

Thessa Fokkema, I. S. Eliëns, Jean-Sébastien Caux

For the one-dimensional repulsive Bose gas (Lieb-Liniger model), we study a special class of highly-excited states obtained by giving a finite momentum to subgroups of particles. These states, which correspond to `splitting' the ground state Fermi sea-like quantum number configuration, are zero-entropy states which display interesting properties more normally associated to ground states. Using a numerically exact method based on integrability, we study these states' excitation spectrum, density correlations and momentum distribution functions. These correlations display power-law asymptotics, and are shown to be accurately described by an effective multicomponent Tomonaga-Luttinger liquid theory whose parameters are obtained from Bethe Ansatz. The non-universal correlation prefactors are moreover obtained from integrability, yielding a completely parameter-free fit of the correlator asymptotics.

arXiv:1401.6857•Physical Review A•Mar 31, 2014•27 citations

The 1D interacting Bose gas in a hard wall box

Murray T. Batchelor, Xin Guan, N. Oelkers +1 more

We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb– Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem. PACS numbers: 05.30.−d, 67.40.Db, 05.30.Jp (Some figures in this article are in colour only in the electronic version) 1.

•Mar 20, 2010•33 citations

One-body correlations and momentum distributions of trapped one-dimensional Bose gases at finite temperature

Attila Takács, Yicheng Zhang, Pasquale Calabrese +3 more

We introduce a general approximate method for calculating the one-body correlations and the momentum distributions of one-dimensional Bose gases at finite interaction strengths and temperatures trapped in smooth confining potentials. Our method combines asymptotic techniques for the long-distance behavior of the gas (similar to Luttinger liquid theory) with known short-distance expansions. We derive analytical results for the limiting cases of strong and weak interactions and provide a general procedure for calculating one-body correlations at any interaction strength. A step-by-step explanation of the numerical method used to compute Green's functions (needed as input to our theory) is included. We benchmark our method against exact numerical calculations and compare its predictions to recent experimental results.

arXiv:2409.16929•Physical Review A•Mar 20, 2025•1 citation
Building on quantum integrability, the authors address the finite temperature one-body dynamical correlation function of 1D Bose gases using an algorithm grounded in the concept of generalized particle-hole excitations relative to a reference state. The correlation function is analyzed from the momentum-energy plane and its line-shape and momentum distribution are presented, well beyond the applicable regime of Tomonaga-Luttinger liquid theory predictions.
The dynamical correlated properties of one-dimensional (1D) Bose gases provide profound understanding of novel physics emergent from collective excitations, for instance, the breakdown of off-diagonal long-range order, and the establishment of Tomonaga-Luttinger liquid theory. However, due to the nonperturbative nature of 1D many-body systems, the exact evaluation of correlation functions is notoriously difficult. Here, by means of a form factor approach based on an algebraic Bethe ansatz and numerics, we present a thorough study on the one-body dynamical correlation function (1BDCF) of the Lieb-Liniger model at finite temperature. The influence of thermal fluctuation and interaction on the behavior of 1BDCF has been demonstrated and analyzed from various perspectives, including the spectral distribution, the line shape at fixed momentum, and the corresponding static correlations.
The one-dimensional Lieb-Liniger Bose gas is a prototypical many-body system featuring universal Tomonaga-Luttinger liquid (TLL) physics and free fermion quantum criticality. We analytically calculate finite temperature local pair correlations for the strong-coupling Bose gas at quantum criticality using the polylog function in the framework of the Yang-Yang thermodynamic equations. We show that the local pair correlation has the universal value ${g}^{(2)}(0)\ensuremath{\approx}2p/(n\ensuremath{\varepsilon})$ in the quantum critical regime, the TLL phase, and the quasiclassical region, where ppp is the pressure per unit length rescaled by the interaction energy $\ensuremath{\varepsilon}=\frac{{\ensuremath{\hbar}}^{2}}{2m}{c}^{2}$ with interaction strength ccc and linear density nnn. This suggests the possibility to test finite temperature local pair correlations for the TLL in the relativistic dispersion regime and to probe quantum criticality with the local correlations beyond the TLL phase. Furthermore, thermodynamic properties at high temperatures are obtained by both high temperature and virial expansion of the Yang-Yang thermodynamic equation.
Exactly solved models provide rigorous understanding of many-body phenomena in strongly correlated systems. In this article, we report a breakthrough in uncovering universal many-body correlated properties of the quantum integrable Lieb-Liniger model. We exactly calculate the dynamical correlation functions by computing the form factors through a newly developed method, by which we are capable of calculating all possible 'relative excitations' over the ground state or a finite temperature state to high precision. Consequently, full spectral functions obtained for the model manifest the unique power-law singularity behaviour at the spectral threshold, confirming the validity of nonlinear Luttinger liquid theory. Our method advances the theory of dynamical correlation functions with high precision towards the thermodynamic limit, and is capable of benchmarking experimental observation of such novel correlated properties.
For the one-dimensional repulsive Bose gas (Lieb-Liniger model), we study a special class of highly-excited states obtained by giving a finite momentum to subgroups of particles. These states, which correspond to `splitting' the ground state Fermi sea-like quantum number configuration, are zero-entropy states which display interesting properties more normally associated to ground states. Using a numerically exact method based on integrability, we study these states' excitation spectrum, density correlations and momentum distribution functions. These correlations display power-law asymptotics, and are shown to be accurately described by an effective multicomponent Tomonaga-Luttinger liquid theory whose parameters are obtained from Bethe Ansatz. The non-universal correlation prefactors are moreover obtained from integrability, yielding a completely parameter-free fit of the correlator asymptotics.
We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb– Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem. PACS numbers: 05.30.−d, 67.40.Db, 05.30.Jp (Some figures in this article are in colour only in the electronic version) 1.
We introduce a general approximate method for calculating the one-body correlations and the momentum distributions of one-dimensional Bose gases at finite interaction strengths and temperatures trapped in smooth confining potentials. Our method combines asymptotic techniques for the long-distance behavior of the gas (similar to Luttinger liquid theory) with known short-distance expansions. We derive analytical results for the limiting cases of strong and weak interactions and provide a general procedure for calculating one-body correlations at any interaction strength. A step-by-step explanation of the numerical method used to compute Green's functions (needed as input to our theory) is included. We benchmark our method against exact numerical calculations and compare its predictions to recent experimental results.