Universal Spin Dynamics in Infinite-Temperature One-Dimensional Quantum Magnets

We address the nature of spin dynamics in various integrable and non-integrable, isotropic and anisotropic quantum spin-S chains, beyond the paradigmatic S=1/2 Heisenberg model. In particular, we investigate the algebraic long-time decay ~ t^-1/z of the spin-spin correlation function at infinite temperature, using state-of-the-art simulations based on tensor network methods. We identify three universal regimes for the spin transport, independent of the exact microscopic model: (i) superdiffusive with z=3/2, as in the Kardar-Parisi-Zhang universality class, when the model is integrable with extra symmetries such as spin isotropy that drive the Drude weight to zero, (ii) ballistic with z=1 when the model is integrable with a finite Drude weight, and (iii) diffusive with z=2 with easy-axis anisotropy or without integrability, at variance with previous observations.