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Thermodynamics of the Repulsive Lieb–Liniger Model

Hans-Peter Eckle

Published Jul 25, 2019• 0 citations
Publisher

Abstract

Abstract This chapter discusses how the Bethe ansatz solution of the one-dimensional Bose gas with repulsive δ-function interaction is extended to finite temperatures, the thermody- namic Bethe ansatz. The excitations of this system consist of particle and hole excitations, which can be described by the corresponding densities of Bethe ansatz roots. It shows how these Bethe ansatz root densities are used to define an appropriate expression for the entropy of the system of Bose particles, which is the main ingredient for the extension of the Bethe ansatz method to finite temperature.

DOI
10.1093/oso/9780199678839.003.0017

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Abstract This chapter discusses how the Bethe ansatz solution of the one-dimensional Bose gas with repulsive δ-function interaction is extended to finite temperatures, the thermody- namic Bethe ansatz. The excitations of this system consist of particle and hole excitations, which can be described by the corresponding densities of Bethe ansatz roots. It shows how these Bethe ansatz root densities are used to define an appropriate expression for the entropy of the system of Bose particles, which is the main ingredient for the extension of the Bethe ansatz method to finite temperature.
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 consider the Lieb–Liniger model for a gas of bosonic δ -interacting particles. Using Algebraic Bethe Ansatz results we compute the thermodynamic limit of the form factors of the density operator between finite entropy eigenstates such as finite temperature states or generic non-equilibrium highly excited states. These form factors are crucial building blocks to obtain the thermodynamic exact dynamic correlation functions of such physically relevant states. As a proof of principle we compute an approximated dynamic structure factor by including only the simplest types of particle-hole excitations and show the agreement with known results.
We address the problem of calculating finite-temperature response functions of an experimentally relevant low-dimensional, strongly correlated system: the integrable one-dimensional Bose gas with a repulsive $\ensuremath{\delta}$-function interaction (the Lieb-Liniger model). Focusing on the dynamical density-density function, we present a Bethe ansatz-based method allowing for its accurate evaluation in finite but large systems, over broad ranges of momenta, frequencies, temperatures, and interaction parameters which are difficult to access using other methods. This allows us to quantify the reshaping of the zero-temperature critical behavior by thermal fluctuations, in experimentally accessible regimes.
We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics-thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.
The momentum- and frequency-dependent one-body correlation function of the one-dimensional interacting Bose gas (Lieb–Liniger model) in the repulsive regime is studied using the Algebraic Bethe Ansatz and numerics. We first provide a determinant representation for the field form factor which is well adapted to numerical evaluation. The correlation function is then reconstructed to high accuracy for systems with finite but large numbers of particles, for a wide range of values of the interaction parameter. Our results are extensively discussed, in particular their specialization to the static case.
Abstract. The momentum- and frequency-dependent one-body correlation function of the one-dimensional interacting Bose gas (Lieb-Liniger model) in the repulsive regime is studied using the Algebraic Bethe Ansatz and numerics. We first provide a determinant representation for the field form factor which is welladapted to numerical evaluation. The correlation function is then reconstructed to high accuracy for systems with finite but large numbers of particles, for a wide range of values of the interaction parameter. Our results are extensively discussed, in particular their specialization to the static case. One-particle dynamical correlations in the one-dimensional Bose gas 2 1.