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Correlated quantum many‐particle systems out of equilibrium are of high interest in many fields, including correlated solids, ultracold atoms, or dense plasmas. Accurate theoretical description of these systems is challenging both, conceptionally and with respect to computational resources. A quantum fluctuations approach is recently presented, which is equivalent to the nonequilibrium GW approximation that promises high accuracy at low computational cost. The method exhibits process time scaling that is linear in the number of time steps, like the G1–G2 scheme, however, at a much reduced computer memory cost. In a second publication, this approach is extended to the two‐time exchange–correlation functions and the dynamic density response properties. Herein, the properties of this approach are analyzed in more detail. The physical meaning of the central approximation, the quantum polarization approximation, is established. It is demonstrated that the method is equivalent to the Bethe–Salpeter equation for the two‐time exchange–correlation function when the generalized Kadanoff–Baym ansatz with Hartree–Fock propagators is applied. [ABSTRACT FROM AUTHOR]

The theory of nonequilibrium Green functions (NEGF) has seen a rapid development over the recent three decades. Applications include diverse correlated many‐body systems in and out of equilibrium. Very good agreement with experiments and available exact theoretical results could be demonstrated if the proper selfenergy approximations were used. However, full two‐time NEGF simulations are computationally costly, as they suffer from a cubic scaling of the computation time with the simulation duration. Recently the G1–G2 scheme that exactly reformulates the generalized Kadanoff–Baym ansatz with Hartree–Fock propagators (HF‐GKBA) into time‐local equations is introduced, which achieves time‐linear scaling and allows for a dramatic speedup and extension of the simulations (Schluenzen et al. Phys. Rev. Lett. 2020, 124, 076601). Remarkably, this scaling is achieved quickly, and also for high‐level selfenergies, including the nonequilibrium GW and T‐matrix approximations (Joost et al. Phys. Rev. B 2020, 101, 245101). Even the dynamically screened ladder approximation is now feasible (Joost et al. Phys. Rev. B 2022, 105, 165155), and also applications to electron‐boson systems are demonstrated. Herein, an overview on recent results that are achieved with the G1–G2 scheme is presented. Problems and open questions are discussed and further ideas of how to overcome the current limitations of the scheme and present are presented. The G1–G2 scheme is illustrated by presenting applying it to the excitation dynamics of Hubbard clusters, to optical excitation of graphene, and to charge transfer during stopping of ions by correlated materials. [ABSTRACT FROM AUTHOR]