Time-Linear G1-G2 NEGF Scheme

• The time-linear G1-G2 scheme is defined as a reformulation of NEGF that converts memory-dependent collision integrals into coupled time-local equations for one- and two-particle Green functions.
• It eliminates history-dependent integrals and reduces computational complexity from O(Nt²) or O(Nt³) to linear O(Nt), significantly speeding up simulations.
• The scheme supports advanced selfenergies such as second-order Born, GW, and T-matrix, with applications ranging from Hubbard clusters and graphene to dense plasmas and open quantum systems.

Searching arXiv for the core G1–G2 NEGF papers and related usage to ground the article in the cited literature. The time-linear G1-G2 scheme is a nonequilibrium Green functions formulation in which the Hartree-Fock generalized Kadanoff-Baym ansatz (HF-GKBA) is rewritten as coupled time-local equations for the time-diagonal one-particle Green function $G^<$ and the correlated part of the time-diagonal two-particle Green function $\mathcal{G}$. In this form, history-dependent collision integrals are replaced by ordinary differential equations for equal-time quantities, which removes memory integrals and yields linear scaling with the number of time steps, $\mathcal{O}(N_t)$, instead of the $\mathcal{O}(N_t^2)$ scaling of standard GKBA for the second-order Born approximation and the $\mathcal{O}(N_t^3)$ scaling of full two-time NEGF and of standard treatments of higher-order selfenergies such as $GW$ and $T$-matrix approximations (Joost et al., 2020, Bonitz et al., 2023).

1. Terminological scope and domain of use

In the many-body context, “G1” denotes the time-diagonal component of the single-particle lesser Green function, $G_{ij}^<(t) \equiv G_{ij}^<(t,t)$, and “G2” denotes the correlated part of the time-diagonal two-particle Green function, $\mathcal{G}_{ijkl}(t)$. The scheme was introduced as a reformulation of GKBA-NEGF simulations into coupled time-local equations for these two objects, and it was later extended to advanced selfenergies, embedding selfenergies, homogeneous quantum plasmas, electron-boson systems, and applications including Hubbard clusters, optical excitation of graphene, and charge transfer during stopping of ions by correlated materials (Joost et al., 2020, Balzer et al., 2022, Makait et al., 2023, Bonitz et al., 2023).

The expression “G1-G2” is not unique across arXiv. In asteroid photometry, the $H,G_1,G_2$ function is an IAU-recommended empirical phase function in which $\mathcal{G}$0 governs the opposition effect and $\mathcal{G}$1 the linear part of the phase curve (Milagros et al., 7 Mar 2025). In numerical analysis of gradient flows, “G1” and “G2” denote first-order and second-order time-linear schemes constructed by energy quadratization and algebraically stable Runge-Kutta discretization (Gong et al., 2019). In He II counterflow, G1 and G2 refer to statistically separated particle populations in velocity PDFs (Mastracci et al., 2018). In Galactic Center astronomy, G1 and G2 designate two gas clouds whose orbital relation was modeled with a drag force (Pfuhl et al., 2014). The time-linear G1-G2 scheme discussed here is the NEGF formulation.

Context	Meaning of G1/G2
NEGF many-body dynamics	Equal-time one-particle and correlated two-particle Green functions
Asteroid photometry	Phase-function parameters in the $\mathcal{G}$2 model
Gradient-flow numerics	First-order and second-order linear schemes
He II counterflow	Two particle-velocity groups in PTV/PDF analysis
Galactic Center	Two distinct gas clouds

This terminological overlap matters because the phrase “time-linear” has a precise meaning in the NEGF literature: linear scaling with simulation duration due to the elimination of temporal memory integrals, not merely a linearized time discretization.

2. Mathematical formulation

The basic single-time equation propagated in the scheme is the equation of motion for the time-diagonal lesser Green function,

$\mathcal{G}$3

The collision integral is exactly rewritten as

$\mathcal{G}$4

so that the dynamics no longer depends on a memory kernel but on the instantaneous correlated two-particle object (Bonitz et al., 2023).

The correlated two-particle quantity is defined as

$\mathcal{G}$5

and its equation of motion in the second-order Born approximation has the time-local form

$\mathcal{G}$6

with $\mathcal{G}$7 (Bonitz et al., 2023).

In the detailed derivation for a general basis, the same structure is written as a coupled system:

$\mathcal{G}$8

All equations are local in time (Joost et al., 2020).

Initial correlations can be incorporated by prescribing the initial value of $\mathcal{G}$9 through the initial one-body and two-body density matrices,

$\mathcal{O}(N_t)$0

or, equivalently, by propagating the initial correlated contribution alongside the inhomogeneous source-driven solution (Joost et al., 2020).

These equations show that the scheme is not a closure at the level of $\mathcal{O}(N_t)$1 alone. Its defining step is the promotion of the equal-time two-particle correlation to a propagated dynamical variable.

3. Mechanism of time-linear scaling

The original computational bottleneck in NEGF comes from the Keldysh-Kadanoff-Baym equations, whose direct two-time solution scales as $\mathcal{O}(N_t)$2. Standard GKBA reduces this to $\mathcal{O}(N_t)$3 for the second-order Born approximation but remains cubic for higher-order selfenergies such as $\mathcal{O}(N_t)$4 and $\mathcal{O}(N_t)$5-matrix treatments (Balzer et al., 2022, Bonitz et al., 2023).

The G1-G2 reformulation achieves $\mathcal{O}(N_t)$6 scaling because all information needed at time $\mathcal{O}(N_t)$7 is contained in $\mathcal{O}(N_t)$8 and $\mathcal{O}(N_t)$9, both propagated by first-order differential equations. There is no temporal convolution over the past, no need to reconstruct two-time objects for the collision term, and no memory integral to reevaluate at every step (Joost et al., 2020, Bonitz et al., 2023).

This gain in temporal scaling is exchanged for a basis-space overhead. The dominant additional object is the rank-4 tensor $\mathcal{O}(N_t^2)$0, which scales as $\mathcal{O}(N_t^2)$1 in storage. In the detailed complexity analysis, the per-step basis scaling depends on representation and selfenergy: for a generic basis it can reach $\mathcal{O}(N_t^2)$2 for $\mathcal{O}(N_t^2)$3 and $\mathcal{O}(N_t^2)$4-matrix schemes; for a Hubbard basis it is reduced to $\mathcal{O}(N_t^2)$5 for SOA, $\mathcal{O}(N_t^2)$6, and $\mathcal{O}(N_t^2)$7-matrix; for jellium it can be as low as $\mathcal{O}(N_t^2)$8 for SOA and $\mathcal{O}(N_t^2)$9, with $\mathcal{O}(N_t^3)$0-matrix scaling as $\mathcal{O}(N_t^3)$1. The reported refinement lowered the overhead relative to the original GKBA to not more than an additional factor $\mathcal{O}(N_t^3)$2 (Joost et al., 2020).

Empirically, the time-linear regime is reached quickly. The overview paper reports that G1-G2 outperforms standard GKBA already after a small number of time steps, less than 50, and summarizes speedups of $\mathcal{O}(N_t^3)$3 compared to two-time approaches (Bonitz et al., 2023). This suggests that the reformulation is not only asymptotically advantageous but practically effective in moderate simulation windows.

4. Selfenergy classes and algorithmic extensions

A central feature of the scheme is that time-linear scaling is retained for selfenergies beyond the second-order Born approximation. The derivations and review articles explicitly list support for the second-order Born approximation, $\mathcal{O}(N_t^3)$4, particle-particle and particle-hole $\mathcal{O}(N_t^3)$5-matrix selfenergies, the dynamically screened ladder approximation, and further approximations including TOA and FLEX (Joost et al., 2020, Bonitz et al., 2023).

For the more advanced selfenergies, the G2 equation acquires additional terms. In the review formulation for the dynamically screened ladder/GW class,

$\mathcal{O}(N_t^3)$6

where $\mathcal{O}(N_t^3)$7 is a source term and $\mathcal{O}(N_t^3)$8, $\mathcal{O}(N_t^3)$9, and $GW$0 represent polarization and ladder contributions (Makait et al., 2023). The precise form depends on the chosen selfenergy, but the evolution remains time-local.

The scheme is described as an exact reformulation of HF-GKBA, not an additional approximation beyond HF-GKBA itself (Bonitz et al., 2023). This point is methodologically important: previous validity assessments of HF-GKBA transfer directly to its G1-G2 formulation.

Open-system extensions were developed by incorporating embedding selfenergies into the same time-local architecture. In that setting the collision integral is split into correlation and embedding parts, $GW$1, and the embedding selfenergy is written as

$GW$2

The resulting formulation introduces an additional time-local equation for a system-environment correlation function $GW$3, while retaining the memory-less structure and time-linear scaling (Balzer et al., 2022).

5. Applications and physical reach

The review literature presents applications to excitation dynamics of Hubbard clusters, optical excitation of graphene, charge transfer during stopping of ions by correlated materials, dense plasmas, and electron-boson systems (Bonitz et al., 2023). These examples are not incidental; they illustrate that the formalism can treat lattice systems, momentum-resolved continuum models, and open quantum systems within a unified time-local propagation framework.

A detailed plasma application is the quasi-one-dimensional jellium study, where the G1-G2 scheme was transformed into momentum representation and applied to dense quantum plasma out of equilibrium (Makait et al., 2023). In homogeneous systems the key equations become

$GW$4

with

$GW$5

For the quasi-one-dimensional model, the effective statically screened Coulomb potential is

$GW$6

The study reports feasible runs on standard university clusters for $GW$7 and $GW$8, with examples requiring 14 GB RAM and 20 hrs wall time, and monitored conservation laws yielding relative errors as low as $GW$9 in total energy after 30,000 time steps (Makait et al., 2023).

That same application also exposed a specific physical effect of the quasi-one-dimensional setting: efficient energy exchange occurs only if projectile and target particles have similar velocities, due to the $T$0-function constraint in the linearized, Markovian limit of the collision integral,

$T$1

The result is a strongly reduced phase space for scattering, making velocity overlap critical for thermalization and stopping (Makait et al., 2023).

In the embedding formulation, a representative application is charge transfer between a Hubbard nanocluster and an additional site. For weak coupling, the embedding G1-G2 scheme matches two-time NEGF reference data exceedingly well. For strong coupling, the original embedding approximation can develop unphysical occupancies if the environment is assumed undisturbed; the formulation was therefore extended by adding a G1-G2 equation for the environment, restoring conservation laws and correct dynamics under strong coupling (Balzer et al., 2022).

6. Limitations, numerical issues, and open questions

The main limitation identified across the literature is the storage and propagation of $T$2. In the general discussion, the memory bottleneck is the $T$3 size of the two-particle object. In the quasi-one-dimensional plasma implementation, the main memory bottleneck is storage of the $T$4 tensor, which in 1D scales as $T$5, and in higher dimensions becomes $T$6, making direct simulations beyond quasi-1D currently out of reach on standard hardware (Makait et al., 2023, Bonitz et al., 2023).

Several numerical pathologies are also documented. In homogeneous systems, long-time simulations can exhibit aliasing due to phase accumulation in $T$7, which may suppress relaxation and produce stationary nonphysical solutions. A reported remedy is Lorentzian-damped Hartree-Fock propagators, $T$8, which damp the memory of initial conditions, suppress numerical aliasing, restore proper relaxation, and can keep violations of conservation laws negligible, with energy violations $T$9 (Makait et al., 2023).

The overview article also identifies numerical stability and $G_{ij}^<(t) \equiv G_{ij}^<(t,t)$0-representability issues for strong interactions or advanced selfenergies, including the possibility of negative densities. The remedies discussed there include post-processing purification techniques, enforcing contraction consistency, and higher-fidelity corrections. Further proposed directions include embedding schemes, stochastic or quantum-fluctuation approaches that approximate $G_{ij}^<(t) \equiv G_{ij}^<(t,t)$1 by ensembles of single-particle trajectories, parallelization and symmetry reduction, reduction of basis for uniform systems, and hierarchical or multilayer embedding (Bonitz et al., 2023).

These limitations clarify the present status of the method. The time-linear G1-G2 scheme removes the temporal memory barrier of NEGF, but it does not remove the many-body state-space barrier. A plausible implication is that its long-term development will depend less on temporal integrators than on compressing, approximating, or restructuring the equal-time two-particle sector while preserving the time-local character that makes the method distinctive.