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arXiv:2107.13784 · Physical review. B./Physical review. B

Fine structure of the nonlinear Drude weights in the spin- 12 XXZ chain

Yuhi Tanikawa, Hosho Katsura

Published Nov 15, 2021• 9 citations
arXivHTMLPDF

Abstract

We study nonlinear Drude weights (NLDWs) for the spin-$\frac{1}{2}$ 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 $U(1)$ 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 $n$. Then, we examine the behavior of the NLDWs at the boundary between the two regions and find that they converge for $n=0,1,2\phantom{\rule{4pt}{0ex}}(\mathrm{mod}\phantom{\rule{3.33333pt}{0ex}}4)$, while they show logarithmic divergence for $n=3\phantom{\rule{4pt}{0ex}}(\mathrm{mod}\phantom{\rule{3.33333pt}{0ex}}4)$. Furthermore, we identify particular anisotropies $\mathrm{\ensuremath{\Delta}}=cos(\ensuremath{\pi}r/(r+1))$ ($r=1,2,3,...$) at which the NLDW at any order $n$ converges to a finite value.

DOI
10.1103/physrevb.104.205116

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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 \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ. 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 \ensuremathΔ=cos(\ensuremathπr/(r+1))\mathrm{\ensuremath{\Delta}}=cos(\ensuremath{\pi}r/(r+1))\ensuremathΔ=cos(\ensuremathπr/(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.
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 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 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.
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We revisit the numerical problem of computing the high temperature spin stiffness, or Drude weight, DDD of the spin-1/21/21/2 XXZXXZXXZ chain using exact diagonalization to systematically analyze its dependence on system symmetries and ensemble. Within the canonical ensemble and for states with zero total magnetization, we find DDD vanishes exactly due to spin-inversion symmetry for all but the anisotropies ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}=cos(\ensuremath{\pi}M/N)$ with $N,M\ensuremath{\in}{\mathbb{Z}}^{+}$ coprimes and N>MN>MN>M, provided system sizes $L\ensuremath{\ge}2N$, for which states with different spin-inversion signature become degenerate due to the underlying sl2s{l}_{2}sl2​ loop algebra symmetry. All these loop-algebra degenerate states carry finite currents which we conjecture [based on data from the system sizes and anisotropies ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}$ (with N<L/2N<L/2N<L/2) available to us] to dominate the grand-canonical ensemble evaluation of DDD in the thermodynamic limit. Including a magnetic flux not only breaks spin-inversion in the zero magnetization sector but also lifts the loop-algebra degeneracies in all symmetry sectors---this effect is more pertinent at smaller $\mathrm{\ensuremath{\Delta}}$ due to the larger contributions to DDD coming from the low-magnetization sectors which are more sensitive to the system's symmetries. Thus we generically find a finite DDD for fluxed rings and arbitrary $0<\mathrm{\ensuremath{\Delta}}<1$ in both ensembles. In contrast, at the isotropic point and in the gapped phase ($\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$) DDD is found to vanish in the thermodynamic limit, independent of symmetry or ensemble. Our analysis demonstrates how convergence to the thermodynamic limit within the gapless phase ($\mathrm{\ensuremath{\Delta}}<1$) may be accelerated and the finite-size anomalies overcome: DDD extrapolates nicely in the thermodynamic limit to either the recently computed lower bound or the thermodynamic Bethe ansatz result provided both spin inversion is broken and the additional degeneracies at the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}$ anisotropies are lifted.
We calculate the bulk and boundary parts of the free energy for an open spin-1/2 XXZ -chain in the critical regime by bosonization. We identify the cut-off independent contributions and determine their amplitudes by comparing with Bethe ansatz calculations at zero temperature T . For the bulk part of the free energy we find agreement with Lukyanov's result (1998 Nucl. Phys. B 522 533). In the boundary part we obtain a cut-off independent term which is linear in T and determines the temperature dependence of the boundary susceptibility in the attractive regime for . We further show that at particular anisotropies where contributions from irrelevant operators with different scaling dimensions cross, logarithmic corrections appear. We give explicit formulae for these terms at those anisotropies where they are most important. We verify our results by comparing with extensive numerical calculations based on a numerical solution of the T = 0 Bethe ansatz equations, the finite temperature Bethe ansatz equations in the quantum transfer matrix formalism, and the density matrix renormalization group applied to transfer matrices.