We revisit the numerical problem of computing the high temperature spin stiffness, or Drude weight,
D of the spin-
1/2 XXZ 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
D 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>M, provided system sizes $L\ensuremath{\ge}2N$, for which states with different spin-inversion signature become degenerate due to the underlying
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/2) available to us] to dominate the grand-canonical ensemble evaluation of
D 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
D coming from the low-magnetization sectors which are more sensitive to the system's symmetries. Thus we generically find a finite
D for fluxed rings and arbitrary $0<\mathrm{\ensuremath{\Delta}}<1$ in both ensembles. In contrast, at the isotropic point and in the gapped phase ($\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$)
D 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:
D 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 report a detailed analysis of the Drude weights for both thermal and spin transport in one-dimensional spin-
21 systems by means of exact diagonalization and analytic approaches at finite temperatures. Transport properties are studied first for the integrable
XXZ model and second for various nonintegrable systems such as the dimerized chain, the frustrated chain, and the spin ladder. We compare our results obtained by exact diagonalization and mean-field theory with those of the Bethe ansatz, bosonization, and other numerical studies in the case of the anisotropic Heisenberg model both in the gapless and gapped regime. In particular, we find indications that the Drude weight for spin transport is finite in the thermodynamic limit for the isotropic chain. For the nonintegrable models, a finite-size analysis of the numerical data for the Drude weights is presented, covering the entire parameter space of the dimerized and frustrated chain. We also discuss which conclusions can be drawn from bosonization regarding the question of whether the Drude weights are finite or not. One of our main results is that the Drude weights vanish in the thermodynamic limit for nonintegrable models.
We present a study of the Drude weight
D(T) of the spin-1/2
XXZ chain in the gapless regime. The thermodynamic Bethe ansatz (TBA) is applied in two different ways. In the first application we employ the particle basis of magnons and their bound states. In this case we rederive and considerably extend earlier work in the literature. However, in the course of our investigation we find arguments that cast doubt on the applicability of the TBA in this case. In a second application by use of the spinon and anti-spinon particle basis we obtain completely different results. Only for anisotropy parameter
Δ close to 0 we find that
D(T) is a monotonously decaying function of temperature. For
Δ close to 1 the behaviour is entirely different showing a finite temperature maximum. Also for the isotropic antiferromagnetic chain (
Δ=1) the results for
D(T) are finite for T=0 as well as for
T>0 with an infinite positive slope at T=0.
Using the Bethe ansatz method, the zero frequency contribution (Drude weight) to the spin current correlations is analyzed for the easy plane antiferromagnetic Heisenberg model. The Drude weight is a monotonically decreasing function of temperature for all $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$; it approaches the zero temperature value with a power law and appears to vanish for all finite temperatures at the isotropic $\ensuremath{\Delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ point.
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-
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.
The Drude weight for the spin transport of the spin-1/2
XXZ Heisenberg chain in the critical regime is evaluated exactly for finite temperatures. We combine the thermodynamic Bethe ansatz with the functional relations of type
Y-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.