A preprint reports a mismatch between two ways of describing nonlinear transport in a one-dimensional repulsive Hubbard model. Exact Bethe-ansatz calculations found finite nonlinear Drude weights, or NLDWs, in the tested quarter-filled cases, even though a low-energy calculation predicted that sufficiently high orders should diverge in the thermodynamic limit.
Here, divergence means that the calculated quantity would grow without bound as the system size becomes effectively infinite. The new calculations instead pointed to finite values in the cases examined, including a high-order case with n = 31. The result is evidence against that prediction within the tested range, not a general ruling out of divergences.
How the response was calculated
The study investigates NLDWs using exact Bethe-ansatz calculations and low-energy effective field theory. For the one-dimensional model, the NLDWs were evaluated from derivatives of the ground-state energy with respect to magnetic flux, using a generalized Kohn formula.
The main calculations focus on the zero-magnetization sector, written as M = N/2. The authors varied interaction strength, filling, system size, the order of the Drude weight and magnetic flux, then compared the exact results with predictions from bosonization and Tomonaga-Luttinger-liquid theory, a low-energy description used for interacting one-dimensional systems.
At quarter filling, defined by N = L/2, the authors solved the Bethe-ansatz equations and calculated Drude weights through the seventh order. They also examined higher orders in finite-size tests. The work is based on model ground-state calculations rather than measurements from people, animals or materials.
The dispute starts at quarter filling
The strong-coupling results show a simple pattern: all odd-order NLDWs have the same magnitude, while their signs alternate from one order to the next. Numerical rescaling agreed with the analytical strong-coupling expansion, including its correction in inverse interaction strength, written as 1/U, and its dependence on order.
The disagreement appears in the low-energy treatment of Umklapp terms, a technical part of the effective theory. That calculation predicts a thermodynamic-limit divergence when the order n exceeds 16K − 3, where K is the theory’s TLL parameter. The predicted first divergent order is n = 7 in strong coupling and n = 13 in weak coupling.
The exact finite-size data did not follow that prediction in the tested quarter-filled cases. Instead, the results, including the n = 31 case, followed linear functions of 1/L², where L denotes system size, and indicated finite NLDWs after extrapolation to the thermodynamic limit.
The authors interpret those extrapolations as evidence against the predicted divergences. They suggest that the relevant Umklapp coupling could be absent or very small, or that perturbative higher-order field theory could be limited in this setting. The calculations do not distinguish between those explanations, so the source of the mismatch remains unresolved.
Half filling shows a different finite-size pattern
At half filling, all NLDWs vanish in the thermodynamic limit. Yet finite-size values increase markedly in the weak-coupling regime, showing how strongly the apparent response can depend on the size of the system being calculated.
For odd orders, the authors derive an asymptotic form in which a power-law factor with alternating sign is multiplied by exponential suppression, written as exp[−L/ξ(U)]. The quantity ξ(U) is the correlation length appearing in the calculation. The analytical expression for its inverse agreed closely with the numerical results.
The paper also proposes extending the linear-response hyperscaling form to higher-order NLDWs near the metal-insulator critical point. In this analysis, the results are organized by the ratio of correlation length to system size. For each fixed number of doped holes, data across interaction strengths and system sizes collapsed onto a single curve when plotted against ξ/L, supporting the proposed hyperscaling form.
What the calculations do—and do not—settle
The findings apply to the specific one-dimensional repulsive Hubbard model and to the fillings, sectors, orders, interactions, fluxes and system sizes examined. The finite quarter-filling extrapolations therefore do not establish finite NLDWs for every order or parameter, while the half-filling results describe the calculated thermodynamic and finite-size behavior.
The central question is whether the quarter-filling discrepancy comes from the Umklapp coupling or from the effective theory used to describe it. The preprint does not prove that quarter-filling Umklapp interactions are absent, and its results are not direct experimental evidence.
The document is identified as arXiv:2608.20269v1 and dated 20 August 2026. It reports support from the FoPM/WINGS program at the University of Tokyo for T.I. and JSPS KAKENHI grants JP23K25783 and 23K25790 for H.K.
Paper data and sources
Original title: Nonlinear Drude weight of the one-dimensional Hubbard model
Authors: Tetsuya Iwasaki, Hosho Katsura
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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