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Interacting quantum spin models are remarkably useful for describingdifferent types of physical, chemical, and biological systems. Significantunderstanding of their equilibrium properties has been achieved to date,especially for the case of spin models with short-range couplings. However,progress towards the development of a comparable understanding in long-rangeinteracting models, in particular out-of-equilibrium, remains limited. In arecent work, we proposed a semiclassical numerical method to study spin models,the discrete truncated Wigner approximation (DTWA), and demonstrated itscapability to correctly capture the dynamics of one- and two-point correlationsin one dimensional (1D) systems. Here we go one step forward and use the DTWAmethod to study the dynamics of correlations in 2D systems with many spins anddifferent types of long-range couplings, in regimes where other numericalmethods are generally unreliable. We compute spatial and time-dependentcorrelations for spin-couplings that decay with distance as a power-law anddetermine the velocity at which correlations propagate through the system.Sharp changes in the behavior of those velocities are found as a function ofthe power-law decay exponent. Our predictions are relevant for a broad range ofsystems including solid state materials, atom-photon systems and ultracoldgases of polar molecules, trapped ions, Rydberg, and magnetic atoms. Wevalidate the DTWA predictions for small 2D systems and 1D systems, butultimately, in the spirt of quantum simulation, experiments will be needed toconfirm our predictions for large 2D systems.

Dynamics of correlations in two-dimensional quantum spin models with long-range interactions: A phase-space Monte-Carlo study