Nonlocal emergent hydrodynamics in a long-range quantum spin system

Generic short-range interacting quantum systems with a conserved quantity exhibit universal diffusive transport at late times. We employ nonequilibrium quantum field theory and semiclassical phase-space simulations to show how this universality is replaced by a more general transport process in a long-range XY spin chain at infinite temperature with couplings decaying algebraically with distance as r(-alpha). While diffusion is recovered for alpha > 1.5, longer-ranged couplings with 0.5 < alpha <= 1.5 give rise to effective classical Levy flights, a random walk with step sizes drawn from a distribution with algebraic tails. We find that the space-time-dependent spin density profiles are self-similar, with scaling functions given by the stable symmetric distributions. As a consequence, for 0.5 < alpha <= 1.5, autocorrelations show hydrodynamic tails decaying in time as t(-1/(2 alpha-1)) and linear-response theory breaks down. Our findings can be readily verified with current trapped ion experiments.