  We benchmark many-body perturbation theory by studying neutral, as well as
non-neutral, excitations of finite lattice systems. The neutral excitation
spectra are obtained by time-propagating the Kadanoff-Baym equations in the
Hartree-Fock and second Born approximations. Our method is equivalent to
solving the Bethe-Salpeter equation with a high-level kernel while respecting
self-consistently, which guarantees the fulfillment of a frequency sum rule. As
a result, we find that a time-local method, such as Hartree-Fock, can give
incomplete spectra, while already the second Born, which is the simplest
time-nonlocal approximation, reproduces well most of the additional
excitations, which are characterized as double-excitations.
