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Generalised hydrodynamics predicts universal ballistic transport inintegrable lattice systems when prepared in generic inhomogeneous initialstates. However, the ballistic contribution to transport can vanish in systemswith additional discrete symmetries. Here we perform large scale numericalsimulations of spin dynamics in the anisotropic Heisenberg $XXZ$ spin $1/2$chain starting from an inhomogeneous mixed initial state which is symmetricwith respect to a combination of spin-reversal and spatial reflection. In theisotropic and easy-axis regimes we find non-ballistic spin transport which weanalyse in detail in terms of scaling exponents of the transportedmagnetisation and scaling profiles of the spin density. While in the easy-axisregime we find accurate evidence of normal diffusion, the spin transport in theisotropic case is clearly super-diffusive, with the scaling exponent very closeto $2/3$, but with universal scaling dynamics which obeys the diffusionequation in nonlinearly scaled time.

Spin diffusion from an inhomogeneous quench in an integrable system