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Physical review. B./Physical review. B

Unimportance of memory for the time nonlocal components of the Kadanoff-Baym equations

Cian C. Reeves, Yuanran Zhu, Chao Yang, Vojtěch Vlček

Published Sep 25, 2023• 10 citations
Publisher

Abstract

The generalized Kadanoff-Baym ansatz (GKBA) is an approximation to the Kadanoff-Baym equations (KBEs) that neglects certain memory effects that contribute to the Green's function at nonequal times. Here we present arguments and numerical results to demonstrate the practical insignificance of the quantities neglected when deriving the GKBA at conditions at which KBE and GKBA are appropriate. We provide a mathematical proof that places a scaling bound on the neglected terms, further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We perform calculations in a range of models, including different system sizes and filling fractions, as well as experimentally relevant nonequilibrium excitations. We find that both the GKBA and KBE capture the dynamics of interacting systems with moderate and even strong interactions well. We explicitly compute terms neglected in the GKBA approximation and show, in the scenarios tested here, that they are orders of magnitude smaller than the terms that are accounted for, i.e., they offer only a small correction when included in the full KBEs.

DOI
10.1103/physrevb.108.115152

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The generalized Kadanoff-Baym ansatz (GKBA) is an approximation to the Kadanoff-Baym equations (KBEs) that neglects certain memory effects that contribute to the Green's function at nonequal times. Here we present arguments and numerical results to demonstrate the practical insignificance of the quantities neglected when deriving the GKBA at conditions at which KBE and GKBA are appropriate. We provide a mathematical proof that places a scaling bound on the neglected terms, further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We perform calculations in a range of models, including different system sizes and filling fractions, as well as experimentally relevant nonequilibrium excitations. We find that both the GKBA and KBE capture the dynamics of interacting systems with moderate and even strong interactions well. We explicitly compute terms neglected in the GKBA approximation and show, in the scenarios tested here, that they are orders of magnitude smaller than the terms that are accounted for, i.e., they offer only a small correction when included in the full KBEs.
The generalized Kadanoff-Baym ansatz (GKBA) is an approximation to the Kadanoff-Baym equations (KBE), that neglects certain memory effects that contribute to the Green's function at non-equal times. Here we present arguments and numerical results to demonstrate the practical insignificance of the quantities neglected when deriving the GKBA at conditions at which KBE and GKBA are appropriate. We provide a mathematical proof that places a scaling bound on the neglected terms, further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We perform calculations in a range of models, including different system sizes and filling fractions, as well as experimentally relevant non-equilibrium excitations. We find that both the GKBA and KBE capture the dynamics of interacting systems with moderate and even strong interactions well. We explicitly compute terms neglected in the GKBA approximation and show, in the scenarios tested here, that they are orders of magnitude smaller than the terms that are accounted for, i.e., they offer only a small correction when included in the full Kadanoff-Baym equations.
In the non-equilibrium Green function calculations, the use of the generalized Kadanoff–Baym ansatz (GKBA) allows for a simple approximate reconstruction of the two-time Green function from its time-diagonal value. With this, a drastic reduction of the computational needs is achieved in time-dependent calculations, making longer time propagation possible and more complex systems accessible. This paper gives credit to the GKBA that was introduced 25 years ago. After a detailed derivation of the GKBA, we recall its application to homogeneous systems and show how to extend it to strongly correlated, inhomogeneous systems. As a proof of concept, we present the results for a two-electron quantum well, where the correct treatment of the correlated electron dynamics is crucial for a correct description of the equilibrium and dynamic properties.
Predicting real-time dynamics in correlated systems is demanding: exact two-time Green's function methods are accurate but often too costly, while the Generalized Kadanoff-Baym Ansatz (GKBA) offers time-linear propagation at the risk of uncontrolled behavior. We examine when and why GKBA fails in a minimal yet informative setting, the Holstein dimer that describes electron–phonon coupling. Using a conserving, fully self-consistent electron–phonon self-energy, we map out parameter regions where GKBA dynamics is stable and where it becomes unstable. We trace the onset of these failures to qualitative changes in the model's ground-state solutions obtained from the full nonequilibrium Green's function theory, thereby providing practical stability bounds for GKBA time evolution. We further show that coupling the dimer to electronic leads can damp and, in part, cure these instabilities. The results supply simple diagnostics and guidelines for reliable GKBA simulations of electron–phonon dynamics.
The Keldysh formalism for nonequilibrium Green's functions is a powerful theoretical framework for the description of the electronic structure, spectroscopy, and dynamics of strongly correlated systems. However, the underlying Kadanoff-Baym equations (KBE) for the two-time Keldysh Green's functions involve a memory kernel, which results in a high computational cost for long simulation times tmax{t}_{\text{max}}tmax​, with a cubic scaling of the computation time with tmax{t}_{\text{max}}tmax​. Truncation of the memory kernel can reduce the computational cost to linear scaling with tmax{t}_{\text{max}}tmax​, but the required memory times will depend on the model and the diagrammatic approximation to the self-energy. We explain how a truncation of the memory kernel can be incorporated into the time-propagation algorithm to solve the KBE, and investigate the systematic truncation of the memory kernel for the Hubbard model in different parameter regimes, and for different diagrammatic approximations. The truncation is easier to control within dynamical mean-field solutions, where it is applied to a momentum-independent self-energy. Here, simulation times up to two orders of magnitude longer are accessible both in the weak and strong coupling regime, allowing for a study of long-time phenomena such as the crossover between prethermalization and thermalization dynamics.
ABSTRACT Predicting real‐time dynamics in correlated systems is demanding: exact two‐time Green's function methods are accurate but often too costly, while the Generalized Kadanoff‐Baym Ansatz (GKBA) offers time‐linear propagation at the risk of uncontrolled behavior. We examine when and why GKBA fails in a minimal yet informative setting, the Holstein dimer that describes electron–phonon coupling. Using a conserving, fully self‐consistent electron–phonon self‐energy, we map out parameter regions where GKBA dynamics is stable and where it becomes unstable. We trace the onset of these failures to qualitative changes in the model's ground‐state solutions obtained from the full nonequilibrium Green's function theory, thereby providing practical stability bounds for GKBA time evolution. We further show that coupling the dimer to electronic leads can damp and, in part, cure these instabilities. The results supply simple diagnostics and guidelines for reliable GKBA simulations of electron–phonon dynamics.
Equilibrium and nonequilibrium properties of nonideal many-particle systems are strongly influenced by correlation effects that are well described by generalized quantum kinetic equations, in particular, the Kadanoff-Baym equations (KBE). However, these equations are usually derived under the assumption of the weakening of initial correlations (Bogolyubov's condition) and, therefore, fail to correctly describe the short time behavior. We demonstrate that this assumption is not necessary for the derivation of the KBE. Using functional derivatives techniques, we present a straightforward generalization of the KBE that allows us to include arbitrary initial correlations and that is more general than previous derivations. As a result, an additional collision integral is obtained, which is being damped out after a few collisions. Our results are complemented with numerical investigations showing the effect of initial correlations.