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