Universal Transport Properties of Continuous quantum gases

Abstract

The Drude weight characterizes ballistic transport in quantum many-body systems, yet a comprehensive understanding and exact analytical results for it remain elusive, especially in multi-component quantum gases. In this work, we leverage Generalized Hydrodynamics and the Thermodynamic Bethe Ansatz method to precisely compute the Drude weights of one-dimensional continuous integrable systems, such as the Lieb-Liniger model and the Bose-Fermi mixture model. We establish an exact, universal relationship between components of the Drude weight matrix and fundamental thermodynamic quantities (e.g., particle, enthalpy, and entropy densities) for the constituent particles with distinct statistics undergo dynamic coupling. For both models, we further derive analytical approximations of the Drude weight in distinct physical regimes and identify universal scaling laws for the Drude weight near quantum phase transitions.Finally, to connect theory with experiment, we propose and simulate two feasible measurement protocols--a linear potential quench and a bipartitioning setup-verifying that they can reliably extract the Drude weights. Our results establish a direct link between macroscopic transport phenomena and microscopic quasiparticle structure, furnishing critical theoretical benchmarks for future ultracold atomic gas experiments.

Figures (16)

• Figure 1: $D_{nn}$ (a) and $D_{ne}$ (b) as functions of chemical potential $\mu$ for different temperatures $T$ at $c=1$. In the zero-temperature limit (here $T=0.001$), a sharp quantum phase transition is observed at $\mu=0$, marking the boundary between the vacuum state and the Luttinger liquid regime. At finite temperatures, this non-analytic singularity is thermally broadened into a continuous crossover. This behavior signals the system's entry into the quantum critical region, where the transport properties are governed by the interplay between the thermal energy $k_B T$ and the chemical potential scale $\mu$.
• Figure 2: Ground-state ($T=0.001$) Drude weights as functions of interaction strength $c$ for the Lieb-Liniger model at chemical potential $\mu=1$. (a) $D_{nn}$ and (b) $D_{ne}$. Solid lines (black) are the exact numerical TBA solutions. Dashed lines (red) are the analytical large-$c$ expansions from Eq. \ref{['eq:Dnn_LL_zTiC_M']} and \ref{['eq:Dne_LL_zTiC_M']}.
• Figure 3: Finite-temperature ($T=1$) Drude weights as functions of interaction strength $c$ for the Lieb-Liniger model at the chemical potential $\mu=1$. (a) $D_{nn}$ and (b) $D_{ne}$. Solid lines (black) are the exact numerical TBA solutions. Dashed lines (red) are the analytical large-$c$ expansions, Eqs. \ref{['eq:Dnn_LL_fTiC']} and \ref{['eq:Dne_LL_fTiC']}.
• Figure 4: The scaling behavior of (a) $D_{nn}$ and (b) $D_{ne}$ versus chemical potential $\mu$ for the phase transitoin with $c = \sqrt{10}$. The critical chemical potential for this transition is $\mu_{c} = 0$, as indicated by the dashed vertical line. The analytical results (symbols) based on Eqs. \ref{['eq:Dnn_LL_LJ_M']} and \ref{['eq:Dne_LL_LJ_M']} are in excellent agreement with numerical solutions (lines) of the TBA equations.
• Figure 5: Drude weight landscapes across quantum phases ($c=\sqrt{10}$). The panels display the values of (a) $D_{nn}$, (b) $D_{nm}$, (c) $D_{ne}$, (d) $D_{mm}$ and (e) $D_{me}$ in the chemical potential–effective field $(\mu, \mathcal{H})$ plane. These contour plots illustrate how the magnitude of each transport coefficient varies across the Vacuum, Fermi, Bose, and Bose-Fermi regions, identifying the distinct phase boundaries.