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arXiv:2604.02426 · Physical Review Letters

Transport and Temperature: Exact Spectrum and Drude Weight for the One-Dimensional Infinite- U Hubbard Model

Shuo Liu, Yuhao Ma, Hitesh J. Changlani, Philip W. Phillips, Bogdan Andrei Bernevig

Published Sep 9, 2026• 0 citations
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Abstract

Understanding charge transport in strongly correlated systems remains a central challenge in condensed matter physics, while exact analytical results for finite-temperature transport remain rare. Here, we investigate charge transport in the one-dimensional Hubbard model in the infinite-interaction limit. We first construct the exact and explicit, i.e., beyond Bethe ansatz energy spectrum and eigenstates. Building on these explicit eigenstates, we derive a closed-form analytical expression for the charge Drude weight at arbitrary temperatures in the hole-dilute limit with a fixed number of doped holes, as well as analytical expressions in the low- and high-temperature limits at fixed hole density. We further show that, at low temperatures, the leading correction to the charge Drude weight is linear in T when the number of holes is fixed, whereas it is quadratic in T when the hole density is fixed. This analytically controlled example highlights the importance of identifying the underlying thermodynamic trajectory when interpreting finite-size numerical studies of charge transport in more general models.

DOI
10.1103/2fwz-cw5h

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Understanding charge transport in strongly correlated systems remains a central challenge in condensed matter physics, while exact analytical results for finite-temperature transport remain rare. Here, we investigate charge transport in the one-dimensional Hubbard model in the infinite-interaction limit. We first construct the exact and explicit, i.e., beyond Bethe ansatz energy spectrum and eigenstates. Building on these explicit eigenstates, we derive a closed-form analytical expression for the charge Drude weight at arbitrary temperatures in the hole-dilute limit with a fixed number of doped holes, as well as analytical expressions in the low- and high-temperature limits at fixed hole density. We further show that, at low temperatures, the leading correction to the charge Drude weight is linear in T when the number of holes is fixed, whereas it is quadratic in T when the hole density is fixed. This analytically controlled example highlights the importance of identifying the underlying thermodynamic trajectory when interpreting finite-size numerical studies of charge transport in more general models.
We investigate nonlinear Drude weights (NLDWs) in the one-dimensional repulsive Hubbard model at zero temperature by combining exact Bethe-ansatz calculations with low-energy effective field theory. At quarter filling, we first derive the strong-coupling expansion of the NLDWs and confirm it numerically over a wide range of interaction strength. We then compare the numerical results with the prediction of the Tomonaga-Luttinger liquid (TLL) description including irrelevant perturbations. While band-curvature corrections yield finite contributions to higher-order NLDWs, the Umklapp interaction predicts divergent NLDWs when the order nnn of the Drude weight exceeds a threshold determined by the TLL parameter. In contrast, finite-size scaling of the exact Bethe-ansatz results indicates that all calculated NLDWs remain finite in the thermodynamic limit, revealing a discrepancy between the exact results and the predictions of the low-energy effective field theory. At half filling, we analyze the finite-size scaling of the NLDWs across the Mott metal-insulator transition. We derive their asymptotic behavior in the insulating phase and propose a hyperscaling ansatz for NLDWs near the critical point, which is verified numerically. Our results clarify the interaction dependence and critical scaling of nonlinear transport coefficients in the one-dimensional Hubbard model and highlight limitations of the conventional low-energy effective description for higher-order transport.
The Drude weight for the one-dimensional Hubbard model is investigated at finite temperatures by using the Bethe ansatz solution. Evaluating finite-size corrections to the thermodynamic Bethe ansatz equations, we obtain the formula for the Drude weight as the response of the system to an external gauge potential. We perform low-temperature expansions of the Drude weight in the case of half-filling as well as away from half-filling, which clearly distinguish the Mott-insulating state from the metallic state.
We calculate the charge and spin Drude weight of the one-dimensional extended Hubbard model with on-site repulsion UUU and nearest-neighbor repulsion VVV at quarter filling using the density-matrix renormalization-group method combined with a variational principle. Our numerical results for the Hubbard model (V=0)(V=0)(V=0) agree with exact results obtained from the Bethe ansatz solution. We obtain the contour map for both Drude weights in the UVUVUV-parameter space for repulsive interactions. We find that the charge Drude weight is discontinuous across the Kosterlitz-Thouless transition between the Luttinger liquid and the charge-density-wave insulator, while the spin Drude weight varies smoothly and remains finite in both phases. Our results can be generally understood using bosonization and renormalization-group results. The finite-size scaling of the charge Drude weight is well fitted by a polynomial function of the inverse system size in the metallic region. In the insulating region, we find an exponential decay of the finite-size corrections with the system size and a universal relation between the charge gap ${\ensuremath{\Delta}}_{c}$ and the correlation length $\ensuremath{\xi}$ which controls this exponential decay.
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: DnnD_{nn}Dnn​, DnmD_{nm}Dnm​, DneD_{ne}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 (DmmD_{mm}Dmm​, DmeD_{me}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 DnmD_{nm}Dnm​, DmmD_{mm}Dmm​, and DmeD_{me}Dme​ across the μ-Hμ\text{-}Hμ-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.
The Drude weight (DW) is an essential quantity that characterizes the quantum transport properties of many-body systems. However, a rigorous understanding and exact computation of DWs, particularly for strongly correlated systems with doping, still remain elusive. In this article, taking advantage of the quantum integrability, we calculate exactly the DWs and Seebeck effect for generic filling factor in one-dimensional Fermi-Hubbard model with arbitrary interaction strengths and magnetic fields. We build up its intrinsic connection to the Luttinger parameters and derive universal scaling laws for DWs across phase transitions. Our results provide a deep understanding of mutual influences in transport between the spin and the charge degrees of freedom, showing a counterintuitive subtle spin-charge coupling effect and uncovering the microscopic origin of the (spin) Seebeck effects in thermal conductivity. Finally, we propose an experimental protocol to measure the DWs in ultracold atomic systems.
We study the charge conductivity of the one-dimensional repulsive Hubbard model at finite temperature using the method of dynamical quantum typicality, focusing at half filling. This numerical approach allows us to obtain current autocorrelation functions from systems with as many as 18 sites, way beyond the range of standard exact diagonalization. Our data clearly suggest that the charge Drude weight vanishes with a power law as a function of system size. The low-frequency dependence of the conductivity is consistent with a finite dc value and thus with diffusion, despite large finite-size effects. Furthermore, we consider the mass-imbalanced Hubbard model for which the charge Drude weight decays exponentially with system size, as expected for a nonintegrable model. We analyze the conductivity and diffusion constant as a function of the mass imbalance and we observe that the conductivity of the lighter component decreases exponentially fast with the mass-imbalance ratio. While in the extreme limit of immobile heavy particles, the Falicov-Kimball model, there is an effective Anderson-localization mechanism leading to a vanishing conductivity of the lighter species, we resolve finite conductivities for an inverse mass ratio of $\ensuremath{\eta}\ensuremath{\gtrsim}0.25$.