For a quantum many-body system, the direct population of states of double-excitation character is a clear indication that correlations importantly contribute to its nonequilibrium properties. We analyze such correlation-induced transitions by propagating the nonequilibrium Green's functions in real-time within the second Born approximation. As crucial benchmarks, we compute the absorption spectrum of few electrons confined in quantum wells of different width. Our results include the full two-time solution of the Kadanoff-Baym equations as well as of their time-diagonal limit and are compared to Hartree-Fock and exact diagonalization data.
For a quantum many-body system, the direct population of states of double-excitation character is a clear indication that correlations importantly contribute to its nonequilibrium properties. We analyze such correlation-induced transitions by propagating the nonequilibrium Green's functions in real-time within the second Born approximation. As crucial benchmarks, we compute the absorption spectrum of few electrons confined in quantum wells of different width. Our results include the full two-time solution of the Kadanoff-Baym equations as well as of their time-diagonal limit and are compared to Hartree-Fock and exact diagonalization data.
We benchmark many-body perturbation theory by studying neutral, as well as non-neutral, excitations of finite lattice systems.The neutral excitation spectra are obtained by time-propagating the Kadanoff-Baym equations in the Hartree-Fock and the second Born approximations.Our method is equivalent to solving the Bethe-Salpeter equation with a high-level kernel while respecting self-consistency, which guarantees the fulfillment of a frequency sum rule.As a result, we find that a time-local method, such as Hartree-Fock, can give incomplete spectra, while already the second Born approximation, which is the simplest time-non-local approximation, reproduces well most of the additional excitations, which are characterized as double excitations.
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.
For a minimal Hubbard-type system at different interaction strengths U , we investigate the density-response for an excitation beyond the linear regime using the generalized Kadanoff-Baym ansatz (GKBA) and the second Born (2B) approximation. We find strong correlation features in the response spectra and establish the connection to an involved double excitation process. By comparing approximate and exact Green's function results, we also observe an anomalous U -dependence of the energy of this double excitation in 2B+GKBA. This is in accordance with earlier findings [K. Balzer et al., EPL 98 , 67002 (2012)] on double excitations in quantum wells.
Abstract. For a minimal Hubbard-type system at different interaction strengths U, we investigate the density-response for an excitation beyond the linear regime using the generalized Kadanoff-Baym ansatz (GKBA) and the second Born (2B) approximation. We find strong correlation features in the response spectra and establish the connection to an involved double excitation process. By comparing approximate and exact Green’s function results, we also observe an anomalous U-dependence of the energy of this double excitation in 2B+GKBA. This is in accordance with earlier findings [K. Balzer et al., EPL 98, 67002 (2012)] on double excitations in quantum wells. 1.
The method of nonequilibrium Green functions (NEGFs) constitutes a solid framework to describe closed and open quantum systems. It captures situations far from equilibrium including arbitrary perturbations and correlations - effects higher than first order in the binary interactions. On the contrary, in the absence of external fields, it reduces to the imaginary-time Matsubara formalism. The abilities (and successes) of the method originate from its self-consistency and the conserving character of many-body approximations that are introduced by diagram technique. On this account, NEGFs have, now for a long time, brought insight into the dynamics of nuclear matter, non-ideal quantum gases and (semiconductor) plasmas. However, applications to localized, finite and strongly interacting systems have emerged only within the recent decade. The dynamical properties follow from quantum kinetic equations - the two-time Kadanoff-Baym equations for the one-particle NEGF - the numerical solution of which is the central topic of this thesis. The non-Markovian structure of these equations thereby inhibits an unlimited, gradual propagation of the nonequilibrium Green function: Any self-consistent, future dynamics of the system is tightly linked to the (complete) past by non-local self-energies. Further, the success in obtaining solutions is being diminished if the system under investigation is not isotropic or inhomogeneous and, in addition, the spatial topology (e.g., the charge carrier or spin density) needs to be resolved in time. On top of this, small isolated systems are generally not candidates showing fast decoherence which offers little hope for smooth and well-damped integration kernels. Despite these difficulties, the thesis at hand is devoted to the computation of the two-time NEGF for finite, closed systems of which ensembles of electrons bound by nuclei forming atoms or molecules are the most natural representatives. Here, in the spirit of the well-studied one-dimensional helium atom, the electron dynamics in model atoms and molecules is addressed in ab initio fashion. To render calculations possible in the first place, (i) a novel hybrid representation of the NEGF is introduced based on the conception of finite elements (FE) and the discrete variable representation (DVR), and (ii) a parallel algorithm is developed that allows for an efficient, well scalable distributed-memory computation of the full two-time Green function. The first point attributes plenty of flexibility and makes the method readily adjustable to different Hamiltonians avoiding the need to, by hand or in intricate numerical manner, calculate the whole bunch of matrix elements of one- and two-particle operators. Also, in consequence, it drastically simplifies the evaluation of self-energies. The second point makes the NEGF approach ready for large-scale, high-performance computing facilities, the resources (of computer power: mainly memory!) of which are indispensable for (some of) the calculations presented. The applicability of the developed methods is demonstrated inter alia on the basis of helium, beryllium, hydrogen and lithium hydride modeled in one spatial dimension whereby electron-electron correlations are treated in the second(-order) Born approximation. After a thorough discussion on the self-consistent ground states, the thesis focuses on the electron dynamics in the linear response regime as well as in the presence of strong external fields. Overall, to assess the the performance of the second Born approximation, the results are compared to the Hartree-Fock approximation, on the one hand, and to the exact solution of the time-(in)dependent Schrödinger equation, on the other hand. In addition, aiming at the possibility of extended time-dependent calculations, the behavior of the generalized Kadanoff-Baym ansatz is discussed.
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.