Spatiotemporal dynamics of classical and quantum density profiles in low-dimensional spin systems

We provide a detailed comparison between the dynamics of high-temperature spatiotemporal correlationfunctions in quantum and classical spin models. In the quantum case, our large-scale numerics are based onthe concept of quantum typicality, which exploits the fact that random pure quantum states can faithfullyapproximate ensemble averages, allowing the simulation of spin-1/2 systems with up to 40 lattice sites. Dueto the exponentially growing Hilbert space, we find that for such system sizes even a single random state issufficient to yield results with extremely low noise that is negligible for most practical purposes. In contrast, aclassical analog of typicality is missing. In particular, we demonstrate that to obtain data with a similar level ofnoise in the classical case, extensive averaging over classical trajectories is required, no matter how large thesystem size. Focusing on (quasi-)one-dimensional spin chains and ladders, we find remarkably good agreementbetween quantum and classical dynamics. This applies not only to cases where both the quantum and classicalmodels are nonintegrable but also to cases where the quantum spin-1/2 model is integrable and the correspondingclassical s → ∞ model is not. Our analysis is based on the comparison of space-time profiles of the spin andenergy correlation functions, where the agreement is found to hold not only in the bulk but also in the tails ofthe resulting density distribution. The mean-squared displacement of the density profiles reflects the nature ofemerging hydrodynamics and is found to exhibit similar scaling for quantum and classical models