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.
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 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.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
We consider the spin-
21 Heisenberg chain (XXX model) weakly perturbed away from integrability by an isotropic next-to-nearest neighbor exchange interaction. Recently, it was conjectured that this model possesses an infinite tower of quasiconserved integrals of motion (charges) [D. Kurlov et al., Phys. Rev. B 105, 104302 (2022)]. In this work we first test this conjecture by investigating how the norm of the adiabatic gauge potential (AGP) scales with the system size, which is known to be a remarkably accurate measure of chaos. We find that for the perturbed XXX chain the behavior of the AGP norm corresponds to neither an integrable nor a chaotic regime, which supports the conjectured quasi-integrability of the model. We then prove the conjecture and explicitly construct the infinite set of quasiconserved charges. Our proof relies on the fact that the XXX chain perturbed by next-to-nearest exchange interaction can be viewed as a truncation of an integrable long-range deformation of the Heisenberg spin chain.
Nonergodic dynamical systems display anomalous transport properties. Prominent examples are integrable quantum systems, whose exceptional properties are diverging dc conductivities. In this Letter, we explain the microscopic origin of ideal conductivity by resorting to the thermodynamic particle content of a system. Using group-theoretic arguments we rigorously resolve the long-standing controversy regarding the nature of spin and charge Drude weights in the absence of chemical potentials. In addition, by employing a hydrodynamic description, we devise an efficient computational method to calculate exact Drude weights from the stationary currents generated in an inhomogeneous quench from bipartitioned initial states. We exemplify the method on the anisotropic Heisenberg model at finite temperatures for the entire range of anisotropies, accessing regimes that are out of reach with other approaches. Quite remarkably, spin Drude weight and asymptotic spin current rates reveal a completely discontinuous (fractal) dependence on the anisotropy parameter.