Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2002.12717 · Physical review. B./Physical review. B

G1-G2 scheme: Dramatic acceleration of nonequilibrium Green functions simulations within the Hartree-Fock generalized Kadanoff-Baym ansatz

Jan‐Philip Joost, Niclas Schlünzen, M. Bönitz

Published Jun 1, 2020• 66 citations
arXivHTMLPDF

Abstract

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 ${T}^{3}$ scaling with the simulation duration $T$. Recently, ${T}^{2}$ 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 ${T}^{1}$ 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 ${N}_{b}$ can be improved compared to our first report: The overhead compared to the original GKBA is not more than an additional factor ${N}_{b}$, even for Hubbard systems.

DOI
10.1103/physrevb.101.245101

Related papers

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

M. Bönitz, Jan‐Philip Joost, Christopher Makait +3 more

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.

arXiv:2312.15030•physica status solidi (b)•Apr 25, 2024•15 citations

Achieving the Scaling Limit for Nonequilibrium Green Functions Simulations

Niclas Schlünzen, Jan‐Philip Joost, M. Bönitz

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.

arXiv:1909.11489•Physical Review Letters•Feb 19, 2020•100 citations

Accelerating nonequilibrium Green function simulations with embedding self-energies

Karsten Balzer, Niclas Schlünzen, Hannes Ohldag +2 more

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.

arXiv:2211.09615•Physical review. B./Physical review. B•Apr 25, 2023•9 citations

Hubbard nanoclusters far from equilibrium

S. Hermanns, Niclas Schlünzen, M. Bönitz

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.

arXiv:1402.7300•Physical Review B•Sep 5, 2014•80 citations

Quantum fluctuations approach to the nonequilibrium GW approximation

Erik Schroedter, Jan‐Philip Joost, M. Bönitz

The quantum dynamics of fermionic or bosonic many-body systems following external excitation can be successfully studied using two-time nonequilibrium Green's functions (NEGF) or single-time reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY hierarchy. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. Here, we develop a different approach where, instead of equations of motion for the many-particle NEGF (or density operators), single-time equations for the correlation functions of fluctuations are analyzed. We present a derivation of the first two equations of the alternative hierarchy of fluctuations and discuss possible decoupling approximations. In particular, we derive the polarization approximation (PA) which is shown to be equivalent to the single-time version [following by applying the generalized Kadanoff-Baym ansatz (GKBA)] of the nonequilibrium GW approximation with exchange effects of NEGF theory, for weak coupling. The main advantage of the quantum fluctuations approach is that the standard ensemble average can be replaced by a semiclassical average over different initial realizations, as was demonstrated before by Lacroix and co-workers [see e.g. D. Lacroix et al., Phys. Rev. B, 2014, 90, 125112]. Here, we introduce the stochastic GW (SGW) approximation and the stochastic polarization approximation (SPA) which are demonstrated to be equivalent to the single-time GW approximation without and with exchange, respectively, in the weak coupling limit. In addition to the standard stochastic approach to sample initial configurations, we also present a deterministic approach. Our numerical tests confirm that our approach has the same favorable linear scaling with the computation time as the recently developed G1-G2 scheme [Schluenzen et al., Phys. Rev. Lett., 2020, 124, 076601]. At the same time, the SPA and SGW approaches scale more favorably with the system size than the G1-G2 scheme, allowing to extend nonequilibrium GW calculations to bigger systems.

arXiv:2204.08250•Condensed Matter Physics•Jan 1, 2022•10 citations

Few-particle quantum dynamics–comparing nonequilibrium Green functions with the generalized Kadanoff–Baym ansatz to density operator theory

S Hermanns, K Balzer, M. Bönitz

The nonequilibrium description of quantum systems requires, for more than two or three particles, the use of a reduced description to be numerically tractable. Two possible approaches are based on either reduced density matrices or nonequilibrium Green functions (NEGF). Both concepts are formulated in terms of hierarchies of coupled equations—the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced density operators and the Martin-Schwinger-hierarchy (MS) for the Green functions, respectively. In both cases, similar approximations are introduced to decouple the hierarchy, yet still many questions regarding the correspondence of both approaches remain open. Here we analyze this correspondence by studying the generalized Kadanoff–Baym ansatz (GKBA) that reduces the NEGF to a single-time theory. Starting from the BBGKY-hierarchy we present the approximations that are necessary to recover the GKBA result both, with Hartree-Fock propagators (HF-GKBA) and propagators in second Born approximation. To test the quality of the HF-GKBA, we study the dynamics of a 4-electron Hubbard nanocluster starting from a strong nonequilibrium initial state and compare to exact results and the Wang-Cassing approximation to the BBGKY hierarchy presented recently by Akbari et al. [1].

arXiv:1211.6959•Journal of Physics Conference Series•Mar 27, 2013•23 citations
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{T}^{3}T3 scaling with the simulation duration TTT. Recently, T2{T}^{2}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{T}^{1}T1 scaling, both for SOA and GWGWGW 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 TTT 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{N}_{b}Nb​ can be improved compared to our first report: The overhead compared to the original GKBA is not more than an additional factor Nb{N}_{b}Nb​, even for Hubbard systems.
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 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.
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 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.
The quantum dynamics of fermionic or bosonic many-body systems following external excitation can be successfully studied using two-time nonequilibrium Green's functions (NEGF) or single-time reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY hierarchy. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. Here, we develop a different approach where, instead of equations of motion for the many-particle NEGF (or density operators), single-time equations for the correlation functions of fluctuations are analyzed. We present a derivation of the first two equations of the alternative hierarchy of fluctuations and discuss possible decoupling approximations. In particular, we derive the polarization approximation (PA) which is shown to be equivalent to the single-time version [following by applying the generalized Kadanoff-Baym ansatz (GKBA)] of the nonequilibrium GW approximation with exchange effects of NEGF theory, for weak coupling. The main advantage of the quantum fluctuations approach is that the standard ensemble average can be replaced by a semiclassical average over different initial realizations, as was demonstrated before by Lacroix and co-workers [see e.g. D. Lacroix et al., Phys. Rev. B, 2014, 90, 125112]. Here, we introduce the stochastic GW (SGW) approximation and the stochastic polarization approximation (SPA) which are demonstrated to be equivalent to the single-time GW approximation without and with exchange, respectively, in the weak coupling limit. In addition to the standard stochastic approach to sample initial configurations, we also present a deterministic approach. Our numerical tests confirm that our approach has the same favorable linear scaling with the computation time as the recently developed G1-G2 scheme [Schluenzen et al., Phys. Rev. Lett., 2020, 124, 076601]. At the same time, the SPA and SGW approaches scale more favorably with the system size than the G1-G2 scheme, allowing to extend nonequilibrium GW calculations to bigger systems.
The nonequilibrium description of quantum systems requires, for more than two or three particles, the use of a reduced description to be numerically tractable. Two possible approaches are based on either reduced density matrices or nonequilibrium Green functions (NEGF). Both concepts are formulated in terms of hierarchies of coupled equations—the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced density operators and the Martin-Schwinger-hierarchy (MS) for the Green functions, respectively. In both cases, similar approximations are introduced to decouple the hierarchy, yet still many questions regarding the correspondence of both approaches remain open. Here we analyze this correspondence by studying the generalized Kadanoff–Baym ansatz (GKBA) that reduces the NEGF to a single-time theory. Starting from the BBGKY-hierarchy we present the approximations that are necessary to recover the GKBA result both, with Hartree-Fock propagators (HF-GKBA) and propagators in second Born approximation. To test the quality of the HF-GKBA, we study the dynamics of a 4-electron Hubbard nanocluster starting from a strong nonequilibrium initial state and compare to exact results and the Wang-Cassing approximation to the BBGKY hierarchy presented recently by Akbari et al. [1].