Two-dimensional correlation propagation dynamics with a cluster discrete phase-space method

Nonequilibrium dynamics of highly controlled quantum systems is a challenging issue in statistical physics and quantum many-body physics, relevant to recent experimental developments of analog and digital quantum simulations. In this work, we develop a discrete phase-space approach for general $\mathrm{SU}(N$) spin systems that utilizes cluster mean-field equations, which capture nontrivial quantum correlations inside each cluster, beyond the capability of the standard discrete truncated Wigner approximation for individual classical spins. Our formalism, based on a cluster phase-point operator, enables efficient numerical samplings of cluster phase-space variables, where the total number of noise variables for a direct product state is independent of the specific way in which the entire system is divided into multiple equally sized finite clusters. We numerically demonstrate that the cluster discrete truncated Wigner approximation (C-dTWA) method can reproduce key results in a recent experiment on correlation propagation dynamics in a two-dimensional Bose-Hubbard system. We further compare the results of C-dTWA for clusters of $2\ifmmode\times\else\texttimes\fi{}2$ sites with those from a two-dimensional tensor network method and discuss that both approaches agree very well in the short-time region, where the energy conservation is well maintained in the tensor network simulations. Since we formulate the C-dTWA method in a general form, it has the potential for application to various dynamical problems in isolated and open quantum systems, even in higher dimensions.