It is well established that at zero magnetic field $h=0$, the spin diffusion constant of the spin-$\frac{1}{2}$ XXZ chain is for anisotropy $\mathrm{\Delta }>1$ 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 $\mathrm{\Delta }=1$ and $\mathrm{\Delta }>1$. 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 $\mathrm{\Delta }>1$ 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-$\frac{1}{2}$ XXZ chain to the one-dimensional (1D) lattice spinless fermion model at zero chemical potential $\mu =0$, for $V/J\ge 1$ 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 $\mu =0$. 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-$\frac{1}{2}$ XXZ chain at $h=0$ for anisotropy $\mathrm{\Delta }\ge 1$ and of the charge transport in the 1D lattice spinless fermion model at $\mu =0$ for $V/J\ge 1$.

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• Magnetic anisotropy
• Magnetic coupling
• Magnetic interactions
• Spinless fermions