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Physical review. B./Physical review. B

On the unphysical solutions of the Kadanoff-Baym equations in linear response: Correlation-induced homogeneous density-distribution and attractors

Adrian Stan

Published Jan 8, 2016• 3 citations
Publisher

Abstract

In modern quantum many-body physics, the Kadanoff-Baym equations have become a crucial component in the treatment of strongly and weakly correlated systems far from equilibrium. From the one-particle Green function $G$($t$,${t}^{\ensuremath{'}}$), they allow for the calculation of time-dependent expectation values of all one-particle observables and the total energy. In this work, for isolated Coulomb systems, the numerical behavior of the Kadanoff-Baym equations is investigated. The electron density dynamics are damped to an unphysical homogeneous density distribution, across both the linear and nonlinear response regimes. Unphysical features are shown to exist for $\mathrm{\ensuremath{\Phi}}$-derivable self-energy approximations, such as Hartree-Fock, second-Born, or $G\phantom{\rule{0}{0ex}}W$, in Hubbard and Coulomb systems, irrespective of interaction strength. With this degree of universality, these findings are pertinent to all two-time formalisms, and suggest the need for a different approach to the dynamics of quantum systems.

DOI
10.1103/physrevb.93.041103

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Comment on “On the unphysical solutions of the Kadanoff-Baym equations in linear response: Correlation-induced homogeneous density-distribution and attractors”

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In a recent Rapid Communication [A. Stan, Phys. Rev. B 93, 041103(R) (2016)], the reliability of the Keldysh-Kadanoff-Baym equations (KBE) using correlated self-energy approximations applied to linear and nonlinear response has been questioned. In particular, the existence of a universal attractor has been predicted that would drive the dynamics of any correlated system towards an unphysical homogeneous density distribution regardless of the system type, the interaction, and the many-body approximation. Moreover, it was conjectured that even the mean-field dynamics would be damped. Here, by performing accurate solutions of the KBE for situations studied in that paper, we prove these claims wrong, being caused by numerical inaccuracies.

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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.

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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.

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In modern quantum many-body physics, the Kadanoff-Baym equations have become a crucial component in the treatment of strongly and weakly correlated systems far from equilibrium. From the one-particle Green function GGG(ttt,t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′), they allow for the calculation of time-dependent expectation values of all one-particle observables and the total energy. In this work, for isolated Coulomb systems, the numerical behavior of the Kadanoff-Baym equations is investigated. The electron density dynamics are damped to an unphysical homogeneous density distribution, across both the linear and nonlinear response regimes. Unphysical features are shown to exist for \ensuremathΦ\mathrm{\ensuremath{\Phi}}\ensuremathΦ-derivable self-energy approximations, such as Hartree-Fock, second-Born, or G\phantom{\rule{0}{0ex}}W, in Hubbard and Coulomb systems, irrespective of interaction strength. With this degree of universality, these findings are pertinent to all two-time formalisms, and suggest the need for a different approach to the dynamics of quantum systems.
In a recent Rapid Communication [A. Stan, Phys. Rev. B 93, 041103(R) (2016)], the reliability of the Keldysh-Kadanoff-Baym equations (KBE) using correlated self-energy approximations applied to linear and nonlinear response has been questioned. In particular, the existence of a universal attractor has been predicted that would drive the dynamics of any correlated system towards an unphysical homogeneous density distribution regardless of the system type, the interaction, and the many-body approximation. Moreover, it was conjectured that even the mean-field dynamics would be damped. Here, by performing accurate solutions of the KBE for situations studied in that paper, we prove these claims wrong, being caused by numerical inaccuracies.
We present a Kadanoff-Baym formalism to study time-dependent phenomena for systems of interacting electrons and phonons in the framework of many-body perturbation theory. The formalism takes correctly into account effects of the initial preparation of an equilibrium state and allows for an explicit time-dependence of both the electronic and phononic degrees of freedom. The method is applied to investigate the charge neutral and non-neutral excitation spectra of a homogeneous, two-site, two-electron Holstein model. This is an extension of a previous study of the ground state properties in the Hartree (H), partially self-consistent Born (Gd) and fully self-consistent Born (GD) approximations published in Säkkinen et al. [J. Chem. Phys. 143, 234101 (2015)]. Here, the homogeneous ground state solution is shown to become unstable for a sufficiently strong interaction while a symmetry-broken ground state solution is shown to be stable in the Hartree approximation. Signatures of this instability are observed for the partially self-consistent Born approximation but are not found for the fully self-consistent Born approximation. By understanding the stability properties, we are able to study the linear response regime by calculating the density-density response function by time-propagation. This amounts to a solution of the Bethe-Salpeter equation with a sophisticated kernel. The results indicate that none of the approximations is able to describe the response function during or beyond the bipolaronic crossover for the parameters investigated. Overall, we provide an extensive discussion on when the approximations are valid and how they fail to describe the studied exact properties of the chosen model system.
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.
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.