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We perform a comparative study of exact and approximate time-evolved densities in small Hubbard chains. The approximate densities are obtained via many-body perturbation theory (Hartree-Fock, 2(nd) Born, GWand T-matrix approximations) within the framework of the time-dependent Kadanoff-Baym equations. Benchmarking approximate results against exact ones allows us to address two rather fundamental issues in the non equilibrium dynamics of strongly correlated systems. I) A characterisation of the performance of several standard MBAs in the non-equilibrium regime. Having a definite notion of how good a specific MBA can be is highly relevant to its application to cases (typically, infinite systems) where exact solutions are not available. Our results show that the T-matrix approximation is overall superior to the other MBAs, at all electron densities. II) A scrutiny of the whole idea of Many Body Perturbation Theory in the Kadanoff-Baym sense, when applied to finite systems. The surprising outcome of our study is that during the time evolution, the KBE develop an unphysical steady state solution. This is a genuinely novel feature of the time-dependent KBE, i.e. is not inherited from possible limitations/approximations in the calculation of the initial state. Our extensive numerical characterisation gives robust evidence that the problem occurs in general, whenever MBPT is applied to finite systems, and approximate self energies based upon infinite partial summations are used. We also offer some more conceptual and general consideration on the dependence of this behaviour on the number of particles and system size. This is followed by our conclusions and glimpses of future work

Artificial damping in the Kadanoff-Baym dynamics of small Hubbard chains