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arXiv:2103.05838 · Physical review. B./Physical review. B

Exact results for nonlinear Drude weights in the spin- 12 XXZ chain

Yuhi Tanikawa, Kazuaki Takasan, Hosho Katsura

Published May 27, 2021• 15 citations
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Abstract

While the linear Drude weight in quantum many-body systems has been studied extensively, its nonlinear counterparts -- called nonlinear Drude weights (NLDWs) -- remain largely unexplored. Here, the authors present the first exact results for the NLDWs of the spin-$\frac{1}{2}$ XXZ chain, a paradigmatic quantum many-body system. Their exact results based on the Bethe ansatz reveal that the NLDWs diverge with the system size in certain parameter regions. Also, the analytical expressions for the NLDWs are obtained when they converge.

DOI
10.1103/physrevb.103.l201120

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While the linear Drude weight in quantum many-body systems has been studied extensively, its nonlinear counterparts -- called nonlinear Drude weights (NLDWs) -- remain largely unexplored. Here, the authors present the first exact results for the NLDWs of the spin-12\frac{1}{2}21​ XXZ chain, a paradigmatic quantum many-body system. Their exact results based on the Bethe ansatz reveal that the NLDWs diverge with the system size in certain parameter regions. Also, the analytical expressions for the NLDWs are obtained when they converge.
We study nonlinear Drude weights (NLDWs) for the spin-12\frac{1}{2}21​ XXZ chain in the critical regime at zero temperature. The NLDWs are generalizations of the linear Drude weight. Via the nonlinear extension of the Kohn formula, they can be read off from higher-order finite-size corrections to the ground-state energy in the presence of a U(1)U(1)U(1) magnetic flux. The analysis of the ground-state energy based on the Bethe ansatz reveals that the NLDWs exhibit convergence, power-law, and logarithmic divergence, depending on the anisotropy parameter $\mathrm{\ensuremath{\Delta}}$. We determine the convergent and power-law divergent regions, which depend on the order of the response nnn. Then, we examine the behavior of the NLDWs at the boundary between the two regions and find that they converge for n=0,1,2(mod4)n=0,1,2\phantom{\rule{4pt}{0ex}}(\mathrm{mod}\phantom{\rule{3.33333pt}{0ex}}4)n=0,1,2(mod4), while they show logarithmic divergence for n=3(mod4)n=3\phantom{\rule{4pt}{0ex}}(\mathrm{mod}\phantom{\rule{3.33333pt}{0ex}}4)n=3(mod4). Furthermore, we identify particular anisotropies $\mathrm{\ensuremath{\Delta}}=cos(\ensuremath{\pi}r/(r+1))$ (r=1,2,3,...r=1,2,3,...r=1,2,3,...) at which the NLDW at any order nnn converges to a finite value.
The spin Drude weight of the spin-1/2 XXZXXZXXZ chain: An analytic finite size study
The Drude weight for the spin transport of the spin-1/2 XXZXXZXXZ Heisenberg chain in the critical regime is evaluated exactly for finite temperatures. We combine the thermodynamic Bethe ansatz with the functional relations of type YYY-system satisfied by the row-to-row transfer matrices. This makes it possible to evaluate the asymptotic behavior of the finite temperature spin Drude weight with respect to the system size. As a result, the Drude weight converges to the results obtained by Zotos (Phys. Rev. Lett. 82, 1764 (1999)), however with very slow convergence upon increase of the system size. This strong size dependence may explain that extrapolations from various numerical approaches yield conflicting results.
Abstract Using generalized hydrodynamics (GHD), we exactly evaluate the finite-temperature spin Drude weight at zero magnetic field for the integrable XXZ chain with arbitrary spin and easy-plane anisotropy. First, we construct the fusion hierarchy of the quantum transfer matrices (T-functions) and derive functional relations (T- and Y-systems) satisfied by the T-functions and certain combinations of them (Y-functions). Through analytical arguments, the Y-system is reduced to a set of non-linear integral equations, equivalent to the thermodynamic Bethe ansatz (TBA) equations. Then, employing GHD, we calculate the spin Drude weight at arbitrary finite temperatures. As a result, a characteristic fractal-like structure of the Drude weight is observed at arbitrary spin, similar to the spin-1/2 case. In our approach, the solutions to the TBA equations (i.e. the Y-functions) can be explicitly written in terms of the T-functions, thus allowing for a systematic calculation of the high-temperature limit of the Drude weight.
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.
Nondecaying dc currents in systems showing ballistic transport are characterized by a constant, known as the Drude weight. Here, the authors obtain analytical results for the spin Drude weight in the interacting anisotropic Heisenberg spin-12\frac{1}{2}21​ chain at low temperatures. The obtained analytical formula reveals that the Drude weight for any finite temperature exhibits a nowhere-continuous dependence on the spin anisotropy of the model.