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.
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.
The time evolution in quantum many-body systems after external excitations is attracting high interest in many fields, including dense plasmas, correlated solids, laser-excited materials, or fermionic and bosonic atoms in optical lattices. The theoretical modeling of these processes is challenging, and the only rigorous quantum-dynamics approach that can treat correlated fermions in two and three dimensions is nonequilibrium Green functions (NEGF). However, NEGF simulations are computationally expensive due to their T3 scaling with the simulation duration T. Recently, T2 scaling was achieved with the generalized Kadanoff-Baym ansatz (GKBA), for the second-order Born (SOA) self energy, which has substantially extended the scope of NEGF simulations. In a recent Letter [Schl\"unzen et al., Phys. Rev. Lett. 124, 076601 (2020).] we demonstrated that GKBA-NEGF simulations can be efficiently mapped onto coupled time-local equations for the single-particle and two-particle Green functions on the time diagonal, hence the method has been called the G1-G2 scheme. This allows one to perform the same simulations with order T1 scaling, both for SOA and GW self energies giving rise to a dramatic speedup. Here we present more details on the G1-G2 scheme, including derivations of the basic equations including results for a general basis, for Hubbard systems, and for jellium. Also, we demonstrate how to incorporate initial correlations into the G1-G2 scheme. Further, the derivations are extended to a broader class of self energies, including the T matrix in the particle-particle and particle-hole channels and the dynamically-screened-ladder approximation. Finally, we demonstrate that, for all self energies, the CPU-time scaling of the G1-G2 scheme with the basis dimension Nb can be improved compared to our first report: The overhead compared to the original GKBA is not more than an additional factor Nb, even for Hubbard systems.
Real-time nonequilibrium Green functions (NEGFs) have been very successfully used to simulate the dynamics of correlated many-particle systems far from equilibrium. However, NEGF simulations are computationally expensive since the effort scales cubically with the simulation duration. Recently, we introduced the G1--G2 scheme that allows for a dramatic reduction to time-linear scaling [N. Schl\"unzen et al., Phys. Rev. Lett. 124, 076601 (2020); J.-P. Joost et al., Phys. Rev. B 101, 245101 (2020)]. Here we tackle another problem: the rapid growth of the computational effort with the system size. In many situations where the system of interest is coupled to a bath, to electric contacts, or to similar macroscopic systems for which a microscopic resolution of the electronic properties is not necessary, efficient simplifications are possible. This is achieved by the introduction of an embedding self-energy---a concept that has been successful in standard NEGF simulations. Here, we demonstrate how the embedding concept can be introduced into the G1--G2 scheme, allowing us to drastically accelerate NEGF embedding simulations. The approach is compatible with all advanced self-energies that can be represented by the G1--G2 scheme [as described in J.-P. Joost et al., Phys. Rev. B 105, 165155 (2022)] and retains the memoryless structure of the equations and their time-linear scaling. As a numerical illustration, we investigate the charge transfer between a Hubbard nanocluster and an additional site which is of relevance for the neutralization of ions in matter.
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.
Correlated real-time dynamics in large, spatially inhomogeneous quantum systems remain difficult to access with nonequilibrium many-body methods. Two-time nonequilibrium Green functions (NEGF) retain dynamical correlations but their computational runtime grows cubically with the number of time steps Nt. This scaling bottleneck could recently be overcome by introducing the G1--G2 scheme that is linear in Nt, but requires propagation of a two-particle correlation function and may suffer from numerical instabilities. This has restricted simulations to small systems with Nb∼102 basis states. Here we introduce a quantum-fluctuation formulation of nonequilibrium Green functions, denoted δNEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products, δG^. This guarantees stable dynamics by preserving the positivity of the reduced density matrices, avoids the explicit storage of the two-particle Green function, and reduces the propagation to a finite ensemble of Hartree-Fock-like trajectories. Combined with a stochastic low-rank decomposition of the correlation functions, the method retains time-linear scaling while extending dynamical GW and particle-particle and particle-hole T-matrix simulations to basis sizes of order Nb∼104. We benchmark δNEGF against exact and HF-GKBA results for lattice systems, finding stable correlated dynamics also at strong coupling. We further demonstrate large-scale simulations of diffusion in two-dimensional Hubbard lattices and ultrafast relaxation in graphene nanoribbon heterostructures with long-range Coulomb interactions. These results establish δNEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.
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 [Formula: see text] 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.
The Hubbard model is a prototype for strongly correlated many-particle systems, including electrons in condensed matter and molecules, as well as for fermions or bosons in optical lattices. While the equilibrium properties of these systems have been studied in detail, the nonequilibrium dynamics following a strong nonperturbative excitation only recently came into the focus of experiments and theory. It is of particular interest how the dynamics depend on the coupling strength and on the particle number and whether there exist universal features in the time evolution. Here, we present results for the dynamics of finite Hubbard clusters based on a self-consistent nonequilibrium Green functions (NEGF) approach invoking the generalized Kadanoff-Baym ansatz (GKBA). We discuss the conserving properties of the GKBA with Hartree-Fock propagators in detail and present a generalized form of the energy conservation criterion of Baym and Kadanoff for NEGF. Furthermore, we demonstrate that the HF-GKBA cures some artifacts of prior two-time NEGF simulations. Besides, this approach substantially speeds up the numerical calculations and thus presents the capability to study comparatively large systems and to extend the analysis to long times allowing for an accurate computation of the excitation spectrum via time propagation. Our data obtained within the second Born approximation compare favorably with exact diagonalization results (available for up to 13 particles) and are expected to have predictive capability for substantially larger systems in the weak-coupling limit.