The spin conductivity in the integrable spin-
1∕2 XXZ chain is known to be infinite at finite temperatures
T for anisotropies $\ensuremath{-}1<\ensuremath{\Delta}<1$. Perturbations, which break integrability, e.g., a next-nearest neighbor coupling ${J}^{\ensuremath{'}}$, render the conductivity finite. We construct numerically a nonlocal conserved operator ${J}_{\ensuremath{\parallel}}$ which is responsible for the finite spin Drude weight of the integrable model and calculate its decay rate for small ${J}^{\ensuremath{'}}$. This allows us to obtain a lower bound for the spin conductivity ${\ensuremath{\sigma}}_{s}\ensuremath{\geqslant}c(T)∕{{J}^{\ensuremath{'}}}^{2}$, where
c(T) is finite for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$. We discuss the implication of our result for the general question how nonlocal conservation laws affect transport properties.
We study magnetization transport in anisotropic spin-
21 chains governed by the integrable XXZ model with and without integrability-breaking perturbations at high temperatures $(T\ensuremath{\rightarrow}\ensuremath{\infty})$ using a hybrid approach that combines exact sum rules with judiciously chosen Ans\"atze. In the integrable XXZ model we find (i) superdiffusion at the isotropic (Heisenberg) point, with frequency dependent conductivity ${\ensuremath{\sigma}}^{\ensuremath{'}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)\ensuremath{\sim}{|\ensuremath{\omega}|}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=\ensuremath{-}3/7$ in close numerical agreement with recent
t-DMRG computations; (ii) a continuously drifting exponent from $\ensuremath{\alpha}=\ensuremath{-}{1}^{+}$ in the XY (gapless) limit of the model to $\ensuremath{\alpha}>0$ within the Ising (gapped) regime; and (iii) a diffusion constant saturating in the XY coupling deep in the Ising limit. We consider two kinds of next-nearest-neighbor integrability breaking perturbations---a simple spin-flip term
(J2) and a three-spin assisted variant
(t2), natural in the fermion particle representation of the spin chain. In the first case we discover a remarkable sensitivity of ${\ensuremath{\sigma}}^{\ensuremath{'}}(\ensuremath{\omega})$ to the sign of
J2, with enhanced low frequency spectral weight and a pronounced upward shift in the magnitude of $\ensuremath{\alpha}$ for
J2>0. Perhaps even more surprising, we find subdiffusion $(\ensuremath{\alpha}>0)$ over a range of
J2<0. By contrast, the effects of the ``fermionic'' three-spin perturbation are sign symmetric; this perturbation produces a clearly observable hydrodynamic relaxation. At large strength of the integrability breaking term ${J}_{2}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ the problem is effectively noninteracting (fermions hopping on odd and even sublattices) and we find $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{-}1$ behavior reminiscent of the XY limit of the integrable XXZ chain. Exact diagonalization studies largely corroborate these findings over accessible frequencies.
Motivated by recent investigations of transport properties of strongly correlated 1D models and thermal conductivity measurements of quasi 1D magnetic systems, we present results for the integrable spin-½ XXZ chain. The thermal conductivity κ(ω) of this model has ℜ κ(ω) = δ(ω), i.e. it is infinite for zero frequency ω. The weight of the delta peak is calculated exactly by a lattice path integral formulation. Numerical results for wide ranges of temperature and anisotropy are presented. The low- and high-temperature limits are studied analytically.
The heat conductivity $\ensuremath{\kappa}(T)$ of integrable models, like the one-dimensional spin-
1/2 nearest-neighbor Heisenberg model, is infinite even at finite temperatures as a consequence of the conservation laws associated with integrability. Small perturbations lead to finite but large transport coefficients which we calculate perturbatively using exact diagonalization and moment expansions. We show that there are two different classes of perturbations. While an interchain coupling of strength ${J}_{\ensuremath{\perp}}$ leads to $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}_{\ensuremath{\perp}}^{2}$ as expected from simple golden-rule arguments, we obtain a much larger $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}^{\ensuremath{'}4}$ for a weak next-nearest-neighbor interaction ${J}^{\ensuremath{'}}$. This can be explained by a new approximate conservation law of the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ Heisenberg chain.
We study magnetization transport at infinite temperature in several spin ladder systems as well as in next-nearest-neighbor coupled spin chains. In the integrable ladder considered we analytically show that the transport is ballistic in sectors with nonzero average magnetization, while numerical simulations of a nonequilibrium stationary setting indicate an anomalous transport in the zero-magnetization sector. For other systems, isotropic Heisenberg ladder and spin chains, showing eigenlevel repulsion typical of quantum chaotic systems, numerical simulations indicate diffusive transport.
It is well established that at zero magnetic field
h=0, the spin diffusion constant of the spin-
21 XXZ chain is for anisotropy $\mathrm{\ensuremath{\Delta}}>1$ finite for both low and high temperatures, implying that the type of spin transport is normal diffusive. Although it is expected that this holds for all finite temperatures
T>0, the calculation of the spin diffusion constant for all
T is typically intractable. Here we consider a class of energy eigenstates that exist both for anisotropies $\mathrm{\ensuremath{\Delta}}=1$ and $\mathrm{\ensuremath{\Delta}}>1$. We show that at the isotropic point their contributions are behind the diffusion constant being infinite, spin transport being anomalous superdiffusive for
T>0. That for $\mathrm{\ensuremath{\Delta}}>1$ such states do not contribute to the diffusion constant is shown to imply it is finite, spin transport being normal diffusive for
T>0. By combining the connection through a Jordan-Wigner transformation of the spin-
21 XXZ chain to the one-dimensional (1D) lattice spinless fermion model at zero chemical potential $\ensuremath{\mu}=0$, for $V/J\ensuremath{\ge}1$ with its Bethe-ansatz solution, where
V is the nearest-neighbor Coulomb repulsion and
J=2t is twice the hopping integral
t, in this paper we also address the issue of the
T>0 charge transport of that model at $\ensuremath{\mu}=0$. It is found to be anomalous superdiffusive for
V/J=1 and normal diffusive for
V/J>1. Our results thus open the door to a key advance in the understanding for all finite temperatures
T>0 of the spin transport in the spin-
21 XXZ chain at
h=0 for anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$ and of the charge transport in the 1D lattice spinless fermion model at $\ensuremath{\mu}=0$ for $V/J\ensuremath{\ge}1$.