Based on the method of hydrodynamic projections we derive a concise formula for the Drude weight of the repulsive Lieb-Liniger \delta δ -Bose gas. Our formula contains only quantities which are obtainable from the thermodynamic Bethe ansatz. The Drude weight is an infinite-dimensional matrix, or bilinear functional: it is bilinear in the currents, and each current may refer to a general linear combination of the conserved charges of the model. As a by-product we obtain the dynamical two-point correlation functions involving charge and current densities at small wavelengths and long times, and in addition the scaled covariance matrix of charge transfer. We expect that our formulas extend to other integrable quantum models.
Based on the method of hydrodynamic projections we derive a concise formula for the Drude weight of the repulsive Lieb-Liniger \delta δ -Bose gas. Our formula contains only quantities which are obtainable from the thermodynamic Bethe ansatz. The Drude weight is an infinite-dimensional matrix, or bilinear functional: it is bilinear in the currents, and each current may refer to a general linear combination of the conserved charges of the model. As a by-product we obtain the dynamical two-point correlation functions involving charge and current densities at small wavelengths and long times, and in addition the scaled covariance matrix of charge transfer. We expect that our formulas extend to other integrable quantum models.
The Drude weight characterizes ballistic transport in quantum many-body systems. Although analytical calculations of Drude weights have been extensively studied in integrable models, their direct connections to finite-temperature macroscopic state functions remain unestablished, especially for continuous multicomponent quantum gases. In the present work, we use generalized hydrodynamics and the thermodynamic Bethe ansätz to calculate exactly the Drude weights for one-dimensional continuous integrable systems, including the Lieb-Liniger and Bose-Fermi mixture models. We derive universal exact relations between Drude weight matrix components and key thermodynamic densities (particle density, enthalpy, entropy). Analytic expressions for Drude weight are obtained across different physical regimes, i.e. strong- and weak-coupling regimes in addition to universal scaling laws near the quantum phase transitions. To bridge theory and experiment, we simulate two experimental protocols, linear potential quench and bipartitioning quench, to enable reliable measurements of the Drude weights. Using these protocols, we calculate the charge and energy Drude weight for Lieb-Liniger gas and compare with recent measurements reported in [Science 391, 290 (2026)], showing excellent agreement with particle density and enthalpy, respectively, thus offering deeper physical insights into experimental observations. Our findings directly link ballistic transport properties to thermodynamics, providing rigorous theoretical benchmarks for future ultracold atomic gas experiments.
Owing to the fact that the particle current operator in nonrelativistic gases is proportional to the total momentum operator, the particle transport in such systems is always ballistic and fully characterized by a Drude weight Δ. The Drude weight can be calculated within linear response theory. It is given by the formula Δ=2πD, where D is the density of the gas. This holds in any dimension and for every equilibrium ensemble, in particular for generalized Gibbs ensembles that describe possible equilibrium states of isolated integrable quantum systems. In the canonical ensemble case, the Drude weight can be equivalently obtained from a generalized susceptibility related to the fluctuations of the conserved particle current. Such susceptibility can be rigorously calculated for the integrable Lieb-Liniger Bose gas in any generalized Gibbs ensemble using a generalized Yang-Yang thermodynamic formalism. The resulting expression agrees with a prediction made within the context of generalized hydrodynamics. It also allows us to see explicitly that, within truly generalized Gibbs ensembles, the conductivity related with the particle current is not determined by the corresponding current-current autocorrelation function.
Building on the Bethe ansatz and generalized hydrodynamics, we rigorously establish universal relations between Drude weights governing charge, spin, and energy transport and thermodynamic properties for the one-dimensional repulsive two-component continuum Fermi gas - the paradigmatic integrable Yang-Gaudin model. We analytically derive thermodynamic expressions for charge Drude weights: Dnn, Dnm, Dne (responses of particle, magnetization and energy currents to a chemical potential gradient) equal the particle density, magnetization, and temperature times entropy density, respectively. These relations constitute a hallmark of ballistic transport, and can be generalized to other 1D continuum integrable systems. Furthermore, we investigate spin Drude weight (Dmm, Dme) at zero and low temperatures. They characterize the magnetization and energy current responses to a magnetic field gradient, revealing an essential spin-charge coupling feature in quantum transport. We find that the Drude weights Dnm, Dmm, and Dme across the μ-H plane remarkably map out the zero-temperature phase boundary. This work fills a critical gap in the understanding of transport within integrable quantum gases and furnishes a direct theoretical foundation for future ultracold-atom experiments.
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.
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.
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.