  The Hubbard model is a prototype for strongly correlated many-particle
systems, including electrons in condensed matter and molecules, as well as for
fermions or bosons in optical lattices. While the equilibrium properties of
these systems have been studied in detail, the nonequilibrium dynamics
following a strong non-perturbative excitation only recently came into the
focus of experiments and theory. It is of particular interest how the dynamics
depend on the coupling strength and on the particle number and whether there
exist universal features in the time evolution. Here, we present results for
the dynamics of finite Hubbard clusters based on a selfconsistent
nonequilibrium Green functions (NEGF) approach invoking the generalized
Kadanoff--Baym ansatz (GKBA). We discuss the conserving properties of the GKBA
with Hartree--Fock propagators in detail and present a generalized form of the
energy conservation criterion of Baym and Kadanoff for NEGF. Furthermore, we
demonstrate that the HF-GKBA cures some artifacts of prior two-time NEGF
simulations. Besides, this approach substantially speeds up the numerical
calculations and thus presents the capability to study comparatively large
systems and to extend the analysis to long times allowing for an accurate
computation of the excitation spectrum via time propagation. Our data obtained
within the second Born approximation compares favorably with exact
diagonalization results (available for up to 13 particles) and are expected to
have predictive capability for substantially larger systems in the weak
coupling limit.
