  We present a Gutzwiller von Neumann dynamics (GvND) method for simulating
equilibrium and nonequilibrium phenomena in strongly correlated electron
systems. Our approach is a real-space formulation of the time-dependent
Gutzwiller approximation method. Applying the GvND method to simulate
interaction quenches in the Peierls-Hubbard model, we demonstrate the
amplification of initial inhomogeneities, which in turn results in the collapse
of quench-induced synchronized oscillation. Moreover, we find a dynamical phase
transition separating two quasi-stationary regimes with rather distinct spatial
distributions of physical quantities after the collapsed oscillation. In
particular, in the strong-coupling regime, the system exhibits a dynamic phase
separation in the quasi-stationary state. Our results thus underscore the
importance of spatial fluctuations in the nonequilibrium dynamics of strongly
correlated systems.
