Diffusion matrix and its application to the spin transport for the XXZ spin-1/2 chain in the gapless regime

A general formula of the diffusion matrix has been recently obtained by generalized hydrodynamics (GHD), and it was shown in [1] that the spin diffusion at infinite temperature of the gapped XXZ chain is non-zero at the Ising limit, while it diverges at the isotropic limit. Being inspired by their study, we applied the GHD formula to the XXZ chain in the gapless regime and obtained exactly the spin diffusion constant at infinite temperature in this regime.
As the result, we observe that the spin diffusion in the gapless regime exhibits a fractal structure.

[1] Diffusion in generalized hydrodynamics and quasiparticle scattering (J. D. Nardis, et al., 2019)