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The time evolution in quantum many-body systems after external excitations isattracting high interest in many fields. The theoretical modeling of theseprocesses is challenging, and the only rigorous quantum-dynamics approach thatcan treat correlated fermions in two and three dimensions is nonequilibriumGreen functions (NEGF). However, NEGF simulations are computationally expensivedue to their $T^3$-scaling with the simulation duration $T$. Recently,$T^2$-scaling was achieved with the generalized Kadanoff--Baym ansatz (GKBA),for the second-order Born (SOA) selfenergy, which has substantially extendedthe scope of NEGF simulations. In a recent Letter [Schl\"unzen \textit{et al.},Phys. Rev. Lett. \textbf{124}, 076601 (2020)] we demonstrated that GKBA-NEGFsimulations can be efficiently mapped onto coupled time-local equations for thesingle-particle and two-particle Green functions on the time diagonal, hencethe method has been called G1--G2 scheme. This allows one to perform the samesimulations with order $T^1$-scaling, both for SOA and $GW$ selfenergies givingrise to a dramatic speedup. Here we present more details on the G1--G2 scheme,including derivations of the basic equations including results for a generalbasis, for Hubbard systems and for jellium. Also, we demonstrate how toincorporate initial correlations into the G1--G2 scheme. Further, thederivations are extended to a broader class of selfenergies, including the $T$matrix in the particle--particle and particle--hole channels, and thedynamically screened-ladder approximation. Finally, we demonstrate that, forall selfenergies, the CPU time scaling of the G1--G2 scheme with the basisdimension, $N_b$, can be improved compared to our first report: the overheadcompared to the original GKBA, is not more than an additional factor $N_b$.

The G1--G2 Scheme: Dramatic Acceleration of Nonequilibrium Green Functions Simulations Within the Hartree--Fock-GKBA