Anomalous relaxation and the high-temperature structure factor of XXZ spin chains

We compute the spin structure factor of XXZ spin chains in the Heisenberg and gapped (Ising) regimes in the high-temperature limit for nonzero magnetization, within the framework of generalized hydrodynamics including diffusive corrections. The structure factor shows a hierarchy of timescales in the gapped phase, owing to $s$-spin magnon bound states (`strings’) of various sizes. Although short strings move ballistically, long strings move primarily diffusively as a result of their collisions with short strings. The interplay between these effects gives rise to anomalous power-law decay of the spin structure factor, with continuously varying exponents, at any fixed separation in the late-time limit. We elucidate the crossover to diffusion (in the gapped phase) and to superdiffusion (at the isotropic point) in the half-filling limit. We verify our results via extensive matrix product operator calculations.