Anomalous Diffusion and Superdiffusion in Integrable Spin Chains via a Hard-Rod Gas Mapping

Abstract: We introduce a multi-species generalization of the hard-rod gas in which each species has a distinct effective length, and the repulsive scattering shift is set by the smaller of the two colliding rods. We argue that this model shares key quasiparticle and scattering features with the XXZ spin chain. We show that fixing only the functional decay of bare velocities with rod length is sufficient to reproduce the XXZ spin-transport phase diagram: diffusion (with anomalous fluctuations) in the anisotropic regime and superdiffusion at the isotropic point. We then demonstrate that the statistics of charge transfer differs qualitatively from that of particle trajectories. For long rods, trajectories are Gaussian in the diffusive regime and appear to exhibit KPZ statistics at the isotropic point, providing a direct microscopic signature of KPZ physics in integrable quasiparticle motion. In contrast, charge-transfer fluctuations are anomalous in the anisotropic regime, while they cross over to Gaussian statistics at late times at the isotropic point, reconciling non-Gaussian trajectory fluctuations with Gaussian charge-transfer statistics. Our results establish classical hard-rod dynamics as a minimal yet quantitatively faithful framework for anomalous spin and charge transport in integrable systems, and offer new insight into the origin of KPZ fluctuations in isotropic integrable models.