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

Journal of the Physical Society of Japan

Solutions of the Lieb–Liniger Integral Equation

Miki Wadati

Published Nov 15, 2002• 39 citations
Publisher

Abstract

Proposed is a method of solving the Lieb–Liniger integral equation for a one-dimensional δ-function bose gas at zero temperature. The integral equation reduces to a set of algebraic equations for the coefficients of the series expansion. For strong coupling case, a few coefficients are enough to have a good approximation. On the contrary, all the coefficients are in the same order for weak coupling case. The expansion in λ= c / K , where c is the coupling constant and K is the cutoff momentum, is a formal one in the sense that the coefficients may include the divergent sums. However, the physical quantities such as the ground state energy are shown to be finite in γ= c / ρ, where ρ is the number density, which agree with the results of the Bogoliubov theory.

DOI
10.1143/jpsj.71.2657

Related papers

On the ground state energy of the δ -function Bose gas*

Craig A. Tracy, Harold Widom

Abstract The weak coupling asymptotics, to order ( c / ρ ) 2 , of the ground state energy of the delta-function Bose gas is derived. Here 2 c ≥ 0 is the delta-function potential amplitude and ρ the density of the gas in the thermodynamic limit. The analysis uses the electrostatic interpretation of the Lieb–Liniger integral equation.

arXiv:1601.04677•Journal of Physics A Mathematical and Theoretical•Jun 6, 2016•25 citations

Exact Analysis of δ-Function Attractive Fermions and Repulsive Bosons in One-Dimension

Toshiaki Iida, Miki Wadati

The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.

arXiv:cond-mat/0503033•Journal of the Physical Society of Japan•Jun 1, 2005•45 citations

Ground state energy of the δ -Bose and Fermi gas at weak coupling from double extrapolation

Sylvain Prolhac

Abstract We consider the ground state energy of the Lieb–Liniger gas with δ interaction in the weak coupling regime γ → 0 . For bosons with repulsive interaction, previous studies gave the expansion e B ( γ ) ≃ γ − 4 γ 3 / 2 / 3 π + ( 1 / 6 − 1 / π 2 ) γ 2 . Using a numerical solution of the Lieb–Liniger integral equation discretized with M points and finite strength γ of the interaction, we obtain very accurate numerics for the next orders after extrapolation on M and γ . The coefficient of γ 5 / 2 in the expansion is found to be approximately equal to − 0.001 587 699 865 505 944 989 29 , accurate within all digits shown. This value is supported by a numerical solution of the Bethe equations with N particles, followed by extrapolation on N and γ . It was identified as ( 3 ζ ( 3 ) / 8 − 1 / 2 ) / π 3 by G Lang. The next two coefficients are also guessed from the numerics. For balanced spin 1/2 fermions with attractive interaction, the best result so far for the ground state energy has been

arXiv:1610.08912•Journal of Physics A Mathematical and Theoretical•Feb 3, 2017•25 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

Ground-state energy and excitation spectrum of the Lieb-Liniger model : accurate analytical results and conjectures about the exact solution

Guillaume Lang, F. W. J. Hekking, Anna Minguzzi

We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.

arXiv:1609.08865•SciPost Physics•Jul 7, 2017•59 citations

The 1D interacting Bose gas in a hard wall box

Murray T. Batchelor, X W 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.

arXiv:cond-mat/0505550•Journal of Physics A Mathematical and General•Aug 23, 2005•65 citations
Proposed is a method of solving the Lieb–Liniger integral equation for a one-dimensional δ-function bose gas at zero temperature. The integral equation reduces to a set of algebraic equations for the coefficients of the series expansion. For strong coupling case, a few coefficients are enough to have a good approximation. On the contrary, all the coefficients are in the same order for weak coupling case. The expansion in λ= c / K , where c is the coupling constant and K is the cutoff momentum, is a formal one in the sense that the coefficients may include the divergent sums. However, the physical quantities such as the ground state energy are shown to be finite in γ= c / ρ, where ρ is the number density, which agree with the results of the Bogoliubov theory.
The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.
Abstract We consider the ground state energy of the Lieb–Liniger gas with δ interaction in the weak coupling regime γ → 0 . For bosons with repulsive interaction, previous studies gave the expansion e B ( γ ) ≃ γ − 4 γ 3 / 2 / 3 π + ( 1 / 6 − 1 / π 2 ) γ 2 . Using a numerical solution of the Lieb–Liniger integral equation discretized with M points and finite strength γ of the interaction, we obtain very accurate numerics for the next orders after extrapolation on M and γ . The coefficient of γ 5 / 2 in the expansion is found to be approximately equal to − 0.001 587 699 865 505 944 989 29 , accurate within all digits shown. This value is supported by a numerical solution of the Bethe equations with N particles, followed by extrapolation on N and γ . It was identified as ( 3 ζ ( 3 ) / 8 − 1 / 2 ) / π 3 by G Lang. The next two coefficients are also guessed from the numerics. For balanced spin 1/2 fermions with attractive interaction, the best result so far for the ground state energy has been
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 study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.
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.