The unimportance of memory for the time non-local components of the Kadanoff-Baym equations

The Kadanoff-Baym equations(KBE) offer a theoretically exact approach for propagating Green's functions under the action of a time-dependent Hamiltonian. The dependence of these equations on the time-nonlocal self-energy (corresponding to memory effects) means that the KBEs are prohibitively expensive to solve in most scenarios. The generalized Kadanoff-Baym ansatz in turn neglects certain memory effects and reduces the numerical scaling from cubic to linear in the number of time steps in the propagation. In this talk, we investigate the validity of the approximation made in the derivation of the GKBA. We provide arguments and numerical evidence that the neglected terms are typically orders of magnitude smaller than the terms that are left. Furthermore, we provide a mathematical proof that bounds the neglected terms further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We test our arguments for several models, including different non-equilibrium excitations, filling fractions, system sizes, and different forms of the Hamiltonian with a variable range of interactions. In almost all cases the observation remains the same. For systems that are well captured by a particular self-energy at equilibrium, the quantities derived from the density matrix are well captured by the GKBA.