The dynamics of strongly correlated fermions following an external excitation reveals extremely rich collective quantum effects. Examples are fermionic atoms in optical lattices, electrons in correlated materials, and dense quantum plasmas. Presently, the only quantum-dynamics approach that rigorously describes these processes in two and three dimensions is the nonequilibrium Green functions (NEGF) method. However, NEGF simulations are computationally expensive due to their T^{3} scaling with the simulation duration T. Recently, T^{2} scaling was achieved with the generalized Kadanoff-Baym ansatz (GKBA), for second-order Born (SOA) selfenergies, which has substantially extended the scope of NEGF simulations. Here we demonstrate that GKBA-NEGF simulations can be performed with order T^{1} scaling, both for SOA and GW selfenergies, and point out the remarkable capabilities of this approach.
Simulations of interacting electrons and bosons out of equilibrium, starting from first principles and aiming at realistic multiscale scenarios, is a grand theoretical challenge. Here, using the formalism of nonequilibrium Green's functions and relying in a crucial way on the recently discovered time-linear formulation of the Kadanoff-Baym equations, we present a versatile toolbox for the simulation of correlated electron-boson dynamics. A large class of methods are available, from the Ehrenfest to the dressed
GD for the treatment of electron-boson interactions in combination with perturbative, i.e., Hartree-Fock and second-Born, or nonperturbative, i.e.,
GW and
T matrices either without or with exchange effects, for the treatment of the Coulomb interaction. In all cases the numerical scaling is linear in time and the equations of motion satisfy all fundamental conservation laws.