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arXiv:1808.03520 · The European Physical Journal Special Topics

Initial correlated states for the generalized Kadanoff–Baym Ansatz without adiabatic switching-on of interactions in closed systems

Miroslav Hopjan, Claudio Verdozzi

Published Jan 15, 2019• 18 citations
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Abstract

We reconsider the Generalized Kadanoff–Baym Ansatz (GKBA) approximation for non-equilibrium Green’s functions and extend it to self-consistently define an equilibrium correlated (within GKBA) state in closed systems. The advantage of the proposed prescription is to avoid the preparation of the initial equilibrium correlated state via adiabatic switching-on of the correlations. A simple model system, namely a Hubbard-dimer, is used to illustrate aspects of the computational implementation and performance of the new scheme.

DOI
10.1140/epjst/e2018-800054-3

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We reconsider the Generalized Kadanoff–Baym Ansatz (GKBA) approximation for non-equilibrium Green’s functions and extend it to self-consistently define an equilibrium correlated (within GKBA) state in closed systems. The advantage of the proposed prescription is to avoid the preparation of the initial equilibrium correlated state via adiabatic switching-on of the correlations. A simple model system, namely a Hubbard-dimer, is used to illustrate aspects of the computational implementation and performance of the new scheme.
Within the nonequilibrium Green's function (NEGF) formalism, the generalized Kadanoff-Baym ansatz (GKBA) has stood out as a computationally cheap method to investigate the dynamics of interacting quantum systems driven out of equilibrium. Current implementations of the NEGF-GKBA, however, suffer from a drawback: real-time simulations require noncorrelated states as initial states. Consequently, initial correlations must be built up through an adiabatic switching of the interaction before turning on any external field, a procedure that can be numerically highly expensive. In this work, we extend the NEGF-GKBA to allow for correlated states as initial states. Our scheme makes it possible to efficiently separate the calculation of the initial state from the real-time simulation, thus paving the way for enlarging the class of systems and external drivings accessible by the already successful NEGF-GKBA. We demonstrate the accuracy of the method and its improved performance in a model donor-acceptor dyad driven out of equilibrium by an external laser pulse.
A fast time propagation method for nonequilibrium Green's functions (NEGF) based on the generalized Kadanoff–Baym Ansatz (GKBA) is applied to a lattice system with a symmetry‐broken equilibrium phase, namely an excitonic insulator (EI). The adiabatic preparation of a correlated symmetry‐broken initial state from a Hartree– Fock wave function within GKBA is assessed by comparing with a solution of the imaginary‐time Dyson equation. It is found that it is possible to reach a symmetry‐ broken correlated initial state with nonzero excitonic order parameter by the adiabatic switching (AS) procedure. It is discussed under which circumstances this is possible in practice within reasonably short switching times.
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.
A fast time propagation method for nonequilibrium Green's functions (NEGF) based on the generalized Kadanoff–Baym Ansatz (GKBA) is applied to a lattice system with a symmetry‐broken equilibrium phase, namely an excitonic insulator (EI). The adiabatic preparation of a correlated symmetry‐broken initial state from a Hartree–Fock wave function within GKBA is assessed by comparing with a solution of the imaginary‐time Dyson equation. It is found that it is possible to reach a symmetry‐broken correlated initial state with nonzero excitonic order parameter by the adiabatic switching (AS) procedure. It is discussed under which circumstances this is possible in practice within reasonably short switching times.
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.
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.