Accelerating Nonequilibrium Green Functions Simulations: The G1–G2 Scheme and Beyond

Abstract

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.

Classical and quantum theory of fluctuations for many‐particle systems out of equilibrium

“…Here, we extend the results presented in References [17,18,26,64] and first discuss different sampling methods in Section 5.2. In particular, we consider results for the density on the first site given by…”

“…The existence of this equation was the basis for eliminating two-particle quantities from the numerical propagation scheme the memory consumption of which constitutes the main bottleneck in the G1-G2 scheme. [17] Instead, we developed a semiclassical stochastic approach to sample random realizations of the single-particle quantity, 𝛿 Ĝ → ΔG 𝜆 , which lead to the stochastic polarization approximation (SPA).…”

“…Moreover, a promising application is to spatially uniform systems for which two-particle quantities require very large computer memory which currently limits the G1-G2 scheme to one-dimensional models. [17,66] This includes the uniform electron gas, electron-hole plasmas and dense quantum plasmas. Further, it will be of interest to compute additional quantities such as spin response functions or the momentum distribution and to compare with benchmarks, such as quantum Monte Carlo results.…”

“…Recently linear scaling with N t has become feasible within the G1-G2 scheme, [13,14] which could be demonstrated even for advanced self-energy approximations like GW and the particle-particle and particle-hole T-matrix approximations. Moreover, the full nonequilibrium version of the dynamically screened ladder approximation could be implemented, for lattice models, [15,16] for details of the scheme, see Reference [17]. The advantage of the linear scaling in the G1-G2 scheme comes with a cost: the simultaneous propagation of the time-diagonal single-particle and correlated two-particle Green functions, G 1 (t) and  2 (t), demands a substantial computational effort for computing and storing all matrix elements of  2 .…”