  We identify a class of one-dimensional spin and fermionic lattice models
which display diverging spin and charge diffusion constants, including several
paradigmatic models of exactly solvable strongly correlated many-body dynamics
such as the isotropic Heisenberg spin chains, the Fermi-Hubbard model, and the
t-J model at the integrable point. Using the hydrodynamic transport theory, we
derive an analytic lower bound on the spin and charge diffusion constants by
calculating the curvature of the corresponding Drude weights at half filling,
and demonstrate that for certain lattice models with isotropic interactions
some of the Noether charges exhibit super-diffusive transport at finite
temperature and half filling.
