This article presents an overview on recent progress in the theory of nonequilibrium Green functions (NEGF). We discuss applications of NEGF simulations to describe the femtosecond dynamics of various finite fermionic systems following an excitation out of equilibrium. This includes the expansion dynamics of ultracold atoms in optical lattices following a confinement quench and the excitation of strongly correlated electrons in a solid by the impact of a charged particle. NEGF, presently, are the only ab initio quantum approach that is able to study the dynamics of correlations for long times in two and three dimensions. However, until recently, NEGF simulations have mostly been performed with rather simple selfenergy approximations such as the second-order Born approximation (SOA). While they correctly capture the qualitative trends of the relaxation towards equilibrium, the reliability and accuracy of these NEGF simulations has remained open, for a long time. Here we report on recent tests of NEGF simulations for finite lattice systems against exact-diagonalization and density-matrix-renormalization-group benchmark data. The results confirm the high accuracy and predictive capability of NEGF simulations—provided selfenergies are used that go beyond the SOA and adequately include strong correlation and dynamical-screening effects. With an extended arsenal of selfenergies that can be used effectively, the NEGF approach has the potential of becoming a powerful simulation tool with broad areas of new applications including strongly correlated solids and ultracold atoms. The present review aims at making such applications possible. To this end we present a selfcontained introduction to the theory of NEGF and give an overview on recent numerical applications to compute the ultrafast relaxation dynamics of correlated fermions. In the second part we give a detailed introduction to selfenergies beyond the SOA. Important examples are the third-order approximation, the approximation, the T-matrix approximation and the fluctuating-exchange approximation. We give a comprehensive summary of the explicit selfenergy expressions for a variety of systems of practical relevance, starting from the most general expressions (general basis) and the Feynman diagrams, and including also the important cases of diagonal basis sets, the Hubbard model and the differences occuring for bosons and fermions. With these details, and information on the computational effort and scaling with the basis size and propagation duration, readers will be able to choose the proper basis set and straightforwardly implement and apply advanced selfenergy approximations to a broad class of systems.

Time reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle and not all statistical theories conserve this particular symmetry, most notably hydrodynamic equations and kinetic equations such as the Boltzmann equation. In this article it is shown analytically that quantum kinetic generalizations of the Boltzmann equation that are derived using the nonequilibrium Green functions formalism as well as all approximations that stem from Φ-derivable selfenergies are time reversal invariant.

Time-reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle, and not all statistical theories conserve this particular symmetry-most notably, hydrodynamic equations and kinetic equations such as the Boltzmann equation. Here, we consider quantum kinetic generalizations of the Boltzmann equation using the method of reduced density operators, leading to the quantum generalization of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. We demonstrate that all commonly used approximations, including Vlasov; Hartree-Fock; and the non-Markovian generalizations of the Landau, T-matrix, and Lenard-Balescu equations, are originally time-reversal invariant, and we formulate a general criterion for time reversibility of approximations to the quantum BBGKY hierarchy. Finally, we illustrate, through the example of the Born approximation, how irreversibility is introduced into quantum kinetic theory via the Markov limit, making the connection with the standard Boltzmann equation. This paper is a complement to paper I (Scharnke et al., J. Math. Phys., 2017, 58, 061903), where the time-reversal invariance of quantum kinetic equations was analysed in the frame of the independent non-equilibrium Green functions formalism. KEYWORDSBBGKY-hierarchy, density operators, quantum dynamics, quantum kinetic theory, time reversibility INTRODUCTIONThe time evolution of quantum many-body systems is of great interest currently in many areas of modern physics and chemistry, for example, in the context of laser-matter interaction, non-stationary transport, or dynamics following an interaction or confinement quench. The theoretical concepts used to study these dynamics are fairly broad and include (but are not limited to) wave function-based approaches, density functional theory, and quantum kinetic theory. The latter treats the time dynamics of the Wigner distribution or, more generally, the density matrix and captures the relaxation towards an equilibrium state (see, e.g., Refs. 1-4). The most famous example of a kinetic equation is the Boltzmann equation, along with quantum generalization, but this equation is known to not be applicable to the short-time dynamics. For this reason, generalized quantum kinetic equations were derived that are non-Markovian in nature (e.g., Refs. 1,3,[5][6][7][8][9] and that have a number of remarkable properties, including the conservation of total energy, in contrast to kinetic energy conservation in the Boltzmann equation. It was recently demonstrated that these generalized quantum kinetic equations are well suited to study the relaxation dynamics of weakly and moderately correlated quantum systems, in very good agreement with experiments with ultra-cold atoms (e.g., Refs. 10, 11) and first-principle density matrix renormalization group methods. [12] Contrib. Plasma Phys.