Two-Time Quantum Fluctuations Approach and its Relation to the Bethe--Salpeter Equation

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. We have recently presented a quantum fluctuations approach which is equivalent to the nonequilibrium $GW$ approximation [E. Schroedter \textit{et al.}, Cond. Matt. Phys. \textbf{25}, 23401 (2022)] that promises high accuracy at low computational cost. In a second publication [E. Schroedter \textit{et al.}, Phys. Rev. B \textbf{108}, 205109 (2023)], this approach was extended to the two-time exchange-correlation functions and the density response properties. Here, we analyze the properties of this approach in more detail. We demonstrate 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.