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:1103.6152 · Physical Review A

Efficient thermodynamic description of multicomponent one-dimensional Bose gases

Andreas Klümper, Ovidiu I. Pâţu

Published Nov 21, 2011• 11 citations
arXivHTMLPDF

Abstract

We present a method of obtaining nonlinear integral equations characterizing the thermodynamics of one-dimensional multicomponent gases interacting via a $\ensuremath{\delta}$-function potential. In the case of the repulsive two-component Bose gas we obtain a simple system of two nonlinear integral equations, allowing for an efficient numerical implementation, in contrast with the infinite number of coupled equations obtained by employing the thermodynamic Bethe ansatz. Our technique makes use of the quantum transfer matrix and the fact that, in a certain continuum limit, multicomponent gases can be obtained from appropriate anisotropic spin chains.

DOI
10.1103/physreva.84.051604

Related papers

Thermodynamics of the one-dimensional multicomponent Fermi gas with a delta -function interaction

P. Schlottmann

The author considers a gas of fermions with parabolic dispersion and N spin-components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. He analyses the discrete Bethe ansatz equations and classifies their solutions according to the string hypothesis. The thermodynamic Bethe-ansatz equations are derived for arbitrary band-filling for both repulsive and attractive interaction in terms of the thermodynamic energy potentials for the classes of eigenstates of the Hamiltonian. These equations are then discussed in several limiting cases, e.g. S=1/2, the ground state (T to 0), for vanishing interaction strength, for strong coupling, in the high-temperature limit, and the large-N limit.

Journal of Physics Condensed Matter•Aug 9, 1993•33 citations

Quantum integrable matrix models of spinor Bose gases in one spatial dimension

Hannes Köper, Thomas Gasenzer

Degenerate spinor Bose gases with repulsive density-density interaction and anti-ferromagnetic spin-spin coupling in one spatial dimension are shown to be described by a quantum integrable matrix extension of the nonlinear Schrödinger model, whose fundamental fields are described by an $m\,\times\,n$ matrix of bosonic field operators. The eigenstates of this model are constructed for arbitrarily sized matrix field operators by means of algebraic Bethe-ansatz techniques, and the corresponding Bethe equations governing the spectra of conserved quantities are derived. The approach thus generalizes previously chosen techniques to account for arbitrary spin multiplets and their spin-spin interaction. Focusing on the specific case of the $2\times2$ model, which is shown to correspond to a spin-$1$ Bose gas, a set of integral equations is derived, which describe its equilibrium thermodynamic properties. From these, the ground state phase diagram is computed both, numerically and analytically in the parameter plane spanned by the chemical potential and an external magnetic field. Furthermore, we establish a sufficient condition for interacting bosons in one dimension to obey the Pauli exclusion principle. In particular, we show that in the presence of paired bound states, no two quasiparticle rapidities can coincide, provided that the Lieb parameter satisfies $γ>4/3$.

arXiv:2604.27885•Apr 30, 2026•0 citations

Thermodynamics, density profiles, and correlation functions of the inhomogeneous one-dimensional spinor Bose gas

Ovidiu I. Pâţu, Andreas Klümper

We investigate the finite-temperature properties of the one-dimensional two-component Bose gas (2CBG) with repulsive contact interaction in a harmonic trap. Making use of a lattice embedding for the 2CBG and the quantum transfer matrix, we derive a system of two nonlinear integral equations characterizing the thermodynamics of the uniform system for all values of the relevant parameters: temperature, strength of the interaction, chemical potential, and magnetic field. This system allows for an easy numerical implementation in stark contrast with the infinite number of equations obtained by employing the thermodynamic Bethe ansatz. We use this exact solution coupled with the local density approximation to compute the density profiles and local density correlation function of the inhomogeneous gas for a wide range of coupling strengths and temperatures. Our results show that the polarization in the center of the trap influences heavily the local correlator especially in the experimentally accessible Tonks-Girardeau regime.

arXiv:1508.07758•Physical Review A•Oct 29, 2015•11 citations

Collective dispersion relations for the one-dimensional interacting two-component Bose and Fermi gases

Murray T. Batchelor, Michael Bortz, Xi-Wen Guan +1 more

We investigate the elementary excitations of charge and spin degrees for the 1D interacting two-component Bose and Fermi gases by means of the discrete Bethe ansatz equations. Analytic results in the limiting cases of strong and weak interactions are derived, where the bosons are treated in the repulsive and the fermions in the strongly attractive regime. We confirm and complement results obtained previously from the Bethe ansatz equations in the thermodynamic limit.

arXiv:cond-mat/0512486•Journal of Statistical Mechanics Theory and Experiment•Mar 24, 2006•30 citations

Bose Gas in One Dimension: Lieb–Liniger Model

Hans-Peter Eckle

Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.

•Jul 25, 2019•0 citations

Long-distance behavior of temperature correlation functions in the one-dimensional Bose gas

Karol Kajetan Kozlowski, Jean Michel Maillet, N. A. Slavnov

We describe a Bethe ansatz-based method to derive, starting from a multiple integral representation, the long-distance asymptotic behavior at finite temperature of the density–density correlation function in the interacting one-dimensional Bose gas. We compute the correlation lengths in terms of solutions of non-linear integral equations of the thermodynamic Bethe ansatz type. Finally, we establish a connection between the results obtained in our approach and the correlation lengths stemming from the quantum transfer matrix method.

arXiv:1011.3149•Journal of Statistical Mechanics Theory and Experiment•Mar 22, 2011•56 citations
We present a method of obtaining nonlinear integral equations characterizing the thermodynamics of one-dimensional multicomponent gases interacting via a \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potential. In the case of the repulsive two-component Bose gas we obtain a simple system of two nonlinear integral equations, allowing for an efficient numerical implementation, in contrast with the infinite number of coupled equations obtained by employing the thermodynamic Bethe ansatz. Our technique makes use of the quantum transfer matrix and the fact that, in a certain continuum limit, multicomponent gases can be obtained from appropriate anisotropic spin chains.
The author considers a gas of fermions with parabolic dispersion and N spin-components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. He analyses the discrete Bethe ansatz equations and classifies their solutions according to the string hypothesis. The thermodynamic Bethe-ansatz equations are derived for arbitrary band-filling for both repulsive and attractive interaction in terms of the thermodynamic energy potentials for the classes of eigenstates of the Hamiltonian. These equations are then discussed in several limiting cases, e.g. S=1/2, the ground state (T to 0), for vanishing interaction strength, for strong coupling, in the high-temperature limit, and the large-N limit.
Degenerate spinor Bose gases with repulsive density-density interaction and anti-ferromagnetic spin-spin coupling in one spatial dimension are shown to be described by a quantum integrable matrix extension of the nonlinear Schrödinger model, whose fundamental fields are described by an m × nm\,\times\,nm×n matrix of bosonic field operators. The eigenstates of this model are constructed for arbitrarily sized matrix field operators by means of algebraic Bethe-ansatz techniques, and the corresponding Bethe equations governing the spectra of conserved quantities are derived. The approach thus generalizes previously chosen techniques to account for arbitrary spin multiplets and their spin-spin interaction. Focusing on the specific case of the 2×22\times22×2 model, which is shown to correspond to a spin-111 Bose gas, a set of integral equations is derived, which describe its equilibrium thermodynamic properties. From these, the ground state phase diagram is computed both, numerically and analytically in the parameter plane spanned by the chemical potential and an external magnetic field. Furthermore, we establish a sufficient condition for interacting bosons in one dimension to obey the Pauli exclusion principle. In particular, we show that in the presence of paired bound states, no two quasiparticle rapidities can coincide, provided that the Lieb parameter satisfies γ>4/3γ>4/3γ>4/3.
We investigate the finite-temperature properties of the one-dimensional two-component Bose gas (2CBG) with repulsive contact interaction in a harmonic trap. Making use of a lattice embedding for the 2CBG and the quantum transfer matrix, we derive a system of two nonlinear integral equations characterizing the thermodynamics of the uniform system for all values of the relevant parameters: temperature, strength of the interaction, chemical potential, and magnetic field. This system allows for an easy numerical implementation in stark contrast with the infinite number of equations obtained by employing the thermodynamic Bethe ansatz. We use this exact solution coupled with the local density approximation to compute the density profiles and local density correlation function of the inhomogeneous gas for a wide range of coupling strengths and temperatures. Our results show that the polarization in the center of the trap influences heavily the local correlator especially in the experimentally accessible Tonks-Girardeau regime.
Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
We describe a Bethe ansatz-based method to derive, starting from a multiple integral representation, the long-distance asymptotic behavior at finite temperature of the density–density correlation function in the interacting one-dimensional Bose gas. We compute the correlation lengths in terms of solutions of non-linear integral equations of the thermodynamic Bethe ansatz type. Finally, we establish a connection between the results obtained in our approach and the correlation lengths stemming from the quantum transfer matrix method.