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.
Finite-temperature Drude weight (spin stiffness)
D(T) is evaluated within the anisotropic spin-
1/2 Heisenberg model on a chain using the exact diagonalization for small systems. It is shown that odd-side chains allow for more reliable scaling and results, in particular, if one takes into account corrections due to low-frequency finite-size anomalies. At high
T and zero magnetization
D is shown to scale to zero approaching the isotropic point $\ensuremath{\Delta}=1$. On the other hand, for $\ensuremath{\Delta}>2$ at all magnetizations
D is nearly exhausted with the overlap with the conserved energy current. Results for the
T variation
D(T) are also presented.
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.
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 illustrate how finite-temperature charge and thermal Drude weights of one-dimensional systems can be obtained from the relaxation of initial states featuring global (left–right) gradients in the chemical potential or temperature. The approach is tested for spinless interacting fermions as well as for the Fermi-Hubbard model, and the behavior in the vicinity of special points (such as half filling or isotropic chains) is discussed. We present technical details on how to implement the calculation in practice using the density matrix renormalization group and show that the non-equilibrium dynamics is often less demanding to simulate numerically and features simpler finite-time transients than the corresponding linear response current correlators; thus, new parameter regimes can become accessible. As an application, we determine the thermal Drude weight of the Hubbard model for temperatures T which are an order of magnitude smaller than those reached in the equilibrium approach. This allows us to demonstrate that at low T and half filling, thermal transport is successively governed by spin excitations and described quantitatively by the Bethe ansatz Drude weight of the Heisenberg chain.
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.