Abstract Using the Bethe ansatz method and the thermodynamic Bethe ansatz equations for the higher-spin integrable XXZ chain, the regular zero frequency contribution to the spin current correlation (spin dc conductivity) is analyzed for the spin-1/2 XXZ chain with the anisotropy 0 ⩽ Δ < 1 . In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly evaluate the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit. We find that the high temperature proportionality constant σ 0 reg of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π / cos − 1 Δ ⩾ 2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ = 1 point is exceptional. Close to this point, σ 0 reg slowly increases in proportion to the first power of the one quasi-particle magnetization. On the other hand, σ 0 reg is proportional to the second power of the same when p 0 approaches irrational numbers.
Using the Bethe ansatz method and generalized hydrodynamics (GHD), properties of the spin transport is analyzed for the integrable XXZ chain with the anisotropy 0 ≤ ∆ < 1.Our approach starts from reconstructing the thermodynamic Bethe ansatz (TBA) equations for the integrable XXZ chain with arbitrary spin.We obtain the fusion hierarchy of the quantum transfer matrices (T -functions) in the arbitrary spin case, and we 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 TBA equations.At first, we use the TBA equations for some higher-spin chains in order to examine the regular zero frequency contribution to the spin current correlation (spin dc conductivity) for the spin-1/2 XXZ chain.In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly obtain the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit.We find that the high temperature proportionality constant σ reg 0 of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π/ cos -1 ∆ ≥ 2 with the gap increasing larger than the second power of growing magnetization on one quasiparticle.The isotropic ∆ = 1 point is exceptional.Close to this point, σ reg 0 slowly increases in proportion to the first power of the one quasi-particle magnetization.On the other hand, σ reg 0 is proportional to the second power of the same when p 0 approaches irrational numbers.At second, we examine the finite-temperature spin Drude weight for the integrable XXZ chain with arbitrary spin, where GHD methods are employed.We calculate the spin Drude weight at zero magnetic field, and we find a characteristic fractal-like structure of the Drude weight at arbitrary spin across a broad range of temperatures, 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 hightemperature limit of the Drude weight, as well as the spin dc conductivity in the spin 1/2 case. B Proof of the equivalence between(2.43) and (2.73) C Delivation of the TBA equations for the integrable XXZ chain with arbitrary spin C.1 Free energy . . . . . . . . . .
Based on the ballistic macroscopic fluctuation theory, the integration of the spin correlation function (spin conductivity) is analyzed for the spin-1/2 XXZ chain in the critical regime. In the time when the magnetization of an infinite spin chain fluctuates from an initial state with a wavelength as long as the infinite length
N, the equal-time two-point spin correlation function is scaled up to
O(1/N). In the state where the ballistic spin transport decays at high temperature
T, the diffusive transport remains on a large scale. We show that the spin conductivity is proportional to
1/T in the limit
T→∞ and its high temperature proportionality constant diverges in the case where one-quasiparticle magnetization is infinitely large. This analysis informs that the superdiffusive spin transport is driven by the
1/N-scaled long-range spin correlation and sheds a light on the dynamic scaling in spin transport at the isotropic point.
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.
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 $V/J\ensuremath{\ge}1$.
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.