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arXiv:2212.08711 · Physical review. B./Physical review. B

Drude weights in one-dimensional systems with a single defect

Kazuaki Takasan, Masaki Oshikawa, Haruki Watanabe

Published Feb 21, 2023• 1 citation
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Abstract

Ballistic transport of a quantum system can be characterized by Drude weight, which quantifies the response of the system to a uniform electric field in the infinitely long timescale. The Drude weight is often discussed in terms of the Kohn formula, which gives the Drude weight by the derivative of the energy eigenvalue of a finite-size system with the periodic boundary condition in terms of the Aharonov-Bohm flux. Recently, the Kohn formula is generalized to nonlinear responses. However, the nonlinear Drude weight determined by the Kohn formula often diverges in the thermodynamic limit. In order to elucidate the issue, in this work we examine a simple example of a one-dimensional tight-binding model in the presence of a single defect at zero temperature. We find that its linear and nonlinear Drude weights given by the Kohn formula (i) depend on the Aharonov-Bohm flux and (ii) diverge proportionally to a power of the system size. We argue that the problem can be attributed to different order of limits. The Drude weight according to the Kohn formula (``Kohn-Drude weight'') indicates the response of a finite-size system to an adiabatic insertion of the Aharonov-Bohm flux. While it is a well-defined physical quantity for a finite-size system, its thermodynamic limit does not always describe the ballistic transport of the bulk. The latter should be rather characterized by a ``bulk Drude weight'' defined by taking the thermodynamic limit first before the zero-frequency limit. While the potential issue of the order of limits has been sometimes discussed within the linear response, the discrepancy between the two limits is amplified in nonlinear Drude weights. We demonstrate the importance of the low-energy excitations of $O(1/L)$, which are excluded from the Kohn-Drude weight, in regularizing the bulk Drude weight.

DOI
10.1103/physrevb.107.075141

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Ballistic transport of a quantum system can be characterized by Drude weight, which quantifies the response of the system to a uniform electric field in the infinitely long timescale. The Drude weight is often discussed in terms of the Kohn formula, which gives the Drude weight by the derivative of the energy eigenvalue of a finite-size system with the periodic boundary condition in terms of the Aharonov-Bohm flux. Recently, the Kohn formula is generalized to nonlinear responses. However, the nonlinear Drude weight determined by the Kohn formula often diverges in the thermodynamic limit. In order to elucidate the issue, in this work we examine a simple example of a one-dimensional tight-binding model in the presence of a single defect at zero temperature. We find that its linear and nonlinear Drude weights given by the Kohn formula (i) depend on the Aharonov-Bohm flux and (ii) diverge proportionally to a power of the system size. We argue that the problem can be attributed to different order of limits. The Drude weight according to the Kohn formula (Kohn-Drude weight'') indicates the response of a finite-size system to an adiabatic insertion of the Aharonov-Bohm flux. While it is a well-defined physical quantity for a finite-size system, its thermodynamic limit does not always describe the ballistic transport of the bulk. The latter should be rather characterized by a bulk Drude weight'' defined by taking the thermodynamic limit first before the zero-frequency limit. While the potential issue of the order of limits has been sometimes discussed within the linear response, the discrepancy between the two limits is amplified in nonlinear Drude weights. We demonstrate the importance of the low-energy excitations of O(1/L)O(1/L)O(1/L), which are excluded from the Kohn-Drude weight, in regularizing the bulk Drude weight.
The adiabatic transport properties of U(1) invariant systems are determined by the dependence of the ground state energy on the twisted boundary condition. We examine a one-dimensional tight-binding model in the presence of a single defect and find that the ground state energy of the model shows a universal dependence on the twist angle that can be fully characterized by the transmission coefficient of the scattering by the defect. We identify resulting pathological behaviors of Drude weights in the large system size limit: (i) both the linear and nonlinear Drude weights depend on the twist angle and (ii) the NNN-th order Drude weight diverges proportionally to the (N−1)(N-1)(N−1)-th power of the system size. To clarify the physical implication of the divergence, we simulate the real-time dynamics of the tight-binding model under a static electric field and show that the divergence does not necessarily imply the large current. Furthermore, we address the relation between our results and the boundary conformal field theory.
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.
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.
For finite systems, the real part of the conductivity is usually decomposed as the sum of a zero frequency delta peak and a finite frequency regular part. In studies with periodic boundary conditions, the Drude weight, i.e., the weight of the zero frequency delta peak, is found to be nonzero for integrable systems, even at very high temperatures, whereas it vanishes for generic (nonintegrable) systems. Paradoxically, for systems with open boundary conditions, it can be shown that the coefficient of the zero frequency delta peak is identically zero for any finite system, regardless of its integrability. In order for the Drude weight to be a thermodynamically meaningful quantity, both kinds of boundary conditions should produce the same answer in the thermodynamic limit. We shed light on these issues by using analytical and numerical methods.
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.
The Drude weight characterizes ballistic transport in quantum many-body systems. Although analytical calculations of Drude weights have been extensively studied in integrable models, their direct connections to finite-temperature macroscopic state functions remain unestablished, especially for continuous multicomponent quantum gases. In the present work, we use generalized hydrodynamics and the thermodynamic Bethe ansätz to calculate exactly the Drude weights for one-dimensional continuous integrable systems, including the Lieb-Liniger and Bose-Fermi mixture models. We derive universal exact relations between Drude weight matrix components and key thermodynamic densities (particle density, enthalpy, entropy). Analytic expressions for Drude weight are obtained across different physical regimes, i.e. strong- and weak-coupling regimes in addition to universal scaling laws near the quantum phase transitions. To bridge theory and experiment, we simulate two experimental protocols, linear potential quench and bipartitioning quench, to enable reliable measurements of the Drude weights. Using these protocols, we calculate the charge and energy Drude weight for Lieb-Liniger gas and compare with recent measurements reported in [Science 391, 290 (2026)], showing excellent agreement with particle density and enthalpy, respectively, thus offering deeper physical insights into experimental observations. Our findings directly link ballistic transport properties to thermodynamics, providing rigorous theoretical benchmarks for future ultracold atomic gas experiments.