Electronic spectra provide direct insight into the excitations and correlations of condensed matter systems. Their description requires electron correlations beyond mean field. In equilibrium, the $GW$ approximation has become the method of choice for many materials. Extending this approximation to nonequilibrium, however, is challenging. Full two-time $GW$ simulations within the nonequilibrium Green's functions approach scale at least cubically with propagation time, whereas improved scaling schemes such as time-local (adiabatic) approximations or the generalized Kadanoff--Baym ansatz usually retain only mean-field character in the spectra. The recently introduced real-time Dyson expansion recovers dynamical correlations in the spectrum at time-linear cost, but has so far been restricted to the second-order Born approximation with mean-field reference propagators. Here we formulate and implement the RT-DE for the nonequilibrium $GW$ self-energy with reference propagators of general form whose off-diagonal evolution is time local. In particular, we assess Hartree--Fock propagators, propagators with statically screened exchange with nonequilibrium screening, and a correlated propagator based on the Hartree--Fock GKBA. Benchmarks against exact diagonalization for a driven two-band lattice model with long-range interactions show that the mean-field and statically screened references yield the most accurate spectra, including satellite structures absent at mean-field level, whereas the GKBA-based approach best captures scattering-induced occupation dynamics, but artificially broadens and splits spectral peaks. When applied to large systems that are beyond the reach of exact methods, the scheme resolves the excitonic replica of the valence band and satellites identified as exciton shake-up, as well as their reshaping with increasing excitation density.
Real-time Dyson expansion for the nonequilibrium GW approximation: Correlated spectra and dependence on the reference propagator
Electronic spectra provide direct insight into the excitations and correlations of condensed matter systems. Their description requires electron correlations beyond mean field. In equilibrium, the GW approximation has become the method of choice for many materials. Extending this approximation to nonequilibrium, however, is challenging. Full two-time GW simulations within the nonequilibrium Green's functions approach scale at least cubically with propagation time, whereas improved scaling schemes such as time-local (adiabatic) approximations or the generalized Kadanoff--Baym ansatz usually retain only mean-field character in the spectra. The recently introduced real-time Dyson expansion recovers dynamical correlations in the spectrum at time-linear cost, but has so far been restricted to the second-order Born approximation with mean-field reference propagators. Here we formulate and implement the RT-DE for the nonequilibrium GW self-energy with reference propagators of general form whose off-diagonal evolution is time local. In particular, we assess Hartree--Fock propagators, propagators with statically screened exchange with nonequilibrium screening, and a correlated propagator based on the Hartree--Fock GKBA. Benchmarks against exact diagonalization for a driven two-band lattice model with long-range interactions show that the mean-field and statically screened references yield the most accurate spectra, including satellite structures absent at mean-field level, whereas the GKBA-based approach best captures scattering-induced occupation dynamics, but artificially broadens and splits spectral peaks. When applied to large systems that are beyond the reach of exact methods, the scheme resolves the excitonic replica of the valence band and satellites identified as exciton shake-up, as well as their reshaping with increasing excitation density.
We present a systematic comparison of the real-time Dyson expansion (RTDE) with established non-equilibrium Green's function approaches for simulating driven, interacting quantum systems. Focusing on density matrix dynamics, time-off-diagonal Green's functions, and time-resolved photoemission spectra, we benchmark RTDE against fully self-consistent Kadanoff-Baym equation (KBE) calculations, the generalized Kadanoff-Baym ansatz (GKBA), and exact diagonalization for small systems using second order many-body perturbation theory. Using a driven two-band Hubbard model, we show that mean-field single particle density matrix trajectories provide a reliable baseline for RTDE across a broad range of interaction strengths and excited-carrier populations. Further, RTDE accurately captures correlation effects in the Green's functions, including long-lived oscillations and revivals that are strongly suppressed by the overdamping inherent to self-consistent KBE schemes. As a consequence, RTDE resolves rich non-equilibrium spectral structure in time-resolved photoemission, such as interaction- and population-dependent quasiparticle splittings and bandgap renormalization, which are largely washed out in self-consistent approaches, yet are present in the exact solutions. Our results demonstrate that RTDE bridges the gap between mean-field propagation and full two-time KBE simulations, retaining favorable linear scaling while capturing essential dynamical correlations relevant for ultrafast spectroscopy.
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.
Time-resolved photoemission spectroscopy is the key technique to probe the real-time nonequilibrium dynamics of electronic states. Theoretical predictions of the time dependent spectral function for realistic systems is however, a challenge. Employing the Kadanoff-Baym equations to find this quantity results in a cubic scaling in the total number of time steps, quickly becoming prohibitive and often fail quantitatively and even qualitatively. In comparison, mean-field methods have more favorable numerical scaling both in the number of time steps and in the complexity associated with the cost of evolving for a single time step, however they miss key spectral properties such as emergent spectral features. Here we present a scheme that allows for the inclusion of dynamical correlations to the spectral function while maintaining the same scaling in the number of time steps as for mean-field approaches, while capturing the emergent physics. Further, the scheme can be efficiently implemented on top of equilibrium real-time many-body perturbation theory schemes and codes. We see excellent agreement with exact results for test systems. Furthermore, we exemplify the method on a periodic system and demonstrate clear evidence that our proposed scheme produces complex spectral features including excitonic band replicas, features that are not observed using static mean-field approaches.
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.
Time-resolved photoemission spectroscopy is the key technique to probe the real-time non-equilibrium dynamics of electronic states. Theoretical predictions of the time dependent spectral function for realistic systems is however, a challenge. Employing the Kadanoff-Baym equations to find this quantity results in a cubic scaling in the total number of time steps, quickly becoming prohibitive and often fail quantitatively and even qualitatively. In comparison, mean-field methods have more favorable numerical scaling both in the number of time steps and in the complexity associated with the cost of evolving for a single time step, however they miss key spectral properties such as emergent spectral features. Here we present a scheme that allows for the inclusion of dynamical correlations to the spectral function while maintaining the same scaling in the number of time steps as for mean-field approaches, while capturing the emergent physics. Further, the scheme can be efficiently implemented on top of equilibrium real-time many-body perturbation theory schemes and codes. We see excellent agreement with exact results for test systems. Furthermore we exemplify the method on a periodic system and demonstrate clear evidence that our proposed scheme produces complex spectral features including excitonic band replicas, features that are not observed using static mean-field approaches.
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.