Auxiliary Hamiltonian representation of the nonequilibrium Dyson equation

The nonequilibrium Dyson (or Kadanoff-Baym) equation, which is an equation of motion with long-range memory kernel for real-time Green functions, underlies many numerical approaches based on the Keldysh formalism. In this paper we map the problem of solving the Dyson equation in real-time onto a noninteracting auxiliary Hamiltonian with additional bath degrees of freedom. The solution of the auxiliary model does not require the evaluation of a memory kernel and can thus be implemented in a very memory efficient way. The mapping is derived for a self-energy which is local in space and is thus directly applicable within nonequilibrium dynamical mean-field theory (DMFT). We apply the method to study the interaction quench in the Hubbard model for an optical lattice with a narrow confinement, using inhomogeneous DMFT in combination with second-order weak-coupling perturbation theory. We find that, although the quench excites pronounced density oscillations, signatures of the two-stage relaxation similar to the homogeneous system can be observed by looking at the time-dependent occupations of natural orbitals.Comment: 14 pages, 11 figure

Multiconfiguration time-dependent Hartree impurity solver for nonequilibrium dynamical mean-field theory

Nonequilibrium dynamical mean-field theory (DMFT) solves correlated lattice models by obtaining their local correlation functions from an effective model consisting of a single impurity in a self-consistently determined bath. The recently developed mapping of this impurity problem from the Keldysh time contour onto a time-dependent single-impurity Anderson model (SIAM) [C. Gramsch et al., Phys. Rev. B 88, 235106 (2013)] allows one to use wave function-based methods in the context of nonequilibrium DMFT. Within this mapping, long times in the DMFT simulation become accessible by an increasing number of bath orbitals, which requires efficient representations of the time-dependent SIAM wave function. These can be achieved by the multiconfiguration time-dependent Hartree (MCTDH) method and its multi-layer extensions. We find that MCTDH outperforms exact diagonalization for large baths in which the latter approach is still within reach and allows for the calculation of SIAMs beyond the system size accessible by exact diagonalization. Moreover, we illustrate the computation of the self-consistent two-time impurity Green's function within the MCTDH second quantization representation.Comment: 12 pages, 8 figure

Hamiltonian-based impurity solver for nonequilibrium dynamical mean-field theory

We derive an exact mapping from the action of nonequilibrium dynamical mean-field theory (DMFT) to a single-impurity Anderson model (SIAM) with time-dependent parameters, which can be solved numerically by exact diagonalization. The representability of the nonequilibrium DMFT action by a SIAM is established as a rather general property of nonequilibrium Green functions. We also obtain the nonequilibrium DMFT equations using the cavity method alone. We show how to numerically obtain the SIAM parameters using Cholesky or eigenvector matrix decompositions. As an application, we use a Krylov-based time propagation method to investigate the Hubbard model in which the hopping is switched on, starting from the atomic limit. Possible future developments are discussed.Comment: 24 pages, 11 figure

Lösung der zweizeitigen Kadanoff-Baym Gleichungen: Anwendung auf Modellatome und -moleküle

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.Die Methode der Nichtgleichgewicht-Greenfunktionen (NGGF) stellt ein robustes Verfahren dar, um sowohl abgeschlossene als auch offene Quantensysteme zu beschreiben. Es erlaubt die Betrachtung der Vielteilchendynamik unter dem Einfluss von Korrelationen - Effekten von höherer als zweiter Ordnung in der Wechselwirkung - und beliebiger Störungen fern vom Gleichgewicht. In Abwesenheit externer Felder reduziert sich die Theorie dagegen auf den imaginärzeitlichen Matsubara Formalismus. Die Stärken (und die Erfolge) der Methode liegen in ihrer Selbstkonsistenz und in der Möglichkeit diagrammatische Vielteilchennäherungen einzuführen, die makroskopische Erhaltungssätze erfüllen. Diesen Eigenschaften ist es zu verdanken, dass NGGFen erfolgreich zur Beschreibung von Kernmaterie, nichtidealen Quantengasen und (Halbleiter-)Plasmen verwendet werden. Anwendungen auf endliche, lokalisierte und stark wechselwirkende Systeme sind jedoch erst innerhalb der letzten zehn Jahre erfolgt. Die dynamischen Eigenschaften folgen aus quantenkinetischen Gleichungen - den zweizeitigen Kadanoff-Baym Gleichungen für die Einteilchen-NGGF -, deren numerische Lösung zentraler Gegenstand dieser Arbeit ist. Die nicht-Markovsche Struktur verhindert dabei eine unbegrenzte, schrittweise Propagation der Nichtgleichgewicht-Greenfunktion: Jede selbstkonsistente, zukünftige Entwicklung des Systems ist durch nichtlokale Selbstenergien streng mit seiner (gesamten) Vorgeschichte verknüpft. Weiter erschwert wird die Lösung, wenn das zu untersuchende System nicht isotrop oder inhomogen ist, so dass zusätzlich die räumliche Topologie (z.B. die Ladungs- oder Spindichte) aufgelöst werden muss. Überdies hinaus weisen kleine, isolierte Systeme generell große Kohärenzzeiten auf, was wenig Hoffnung zur Annahme von glatten und deutlich gedämpften Stoßintegralen bietet. Trotz dieser Schwierigkeiten beschäftigt sich die vorliegende Dissertation mit der Berechnung der zweizeitigen NGGF von endlichen, abgeschlossenen Systemen. Im Sinne des wohl untersuchten eindimensionalen Heliumatoms wird die Elektronendynamik in Modellatomen und -molekülen aufgegriffen. Die Beschreibung erfolgt dabei ab initio. Um die Rechnungen überhaupt zu ermöglichen, wird (i) eine neuartige Hybriddarstellung der NGGF angewendet, die auf der Idee von finiten Elementen (FE) und der diskreten Variablendarstellung (DVR) beruht, und (ii) wird ein paralleler Algorithmus entwickelt, der eine effiziente, gut skalierbare Berechnung der vollen zweizeitigen Greenfunktion unter Berücksichtigung von verteiltem Speicher erlaubt. Die FE-DVR-Darstellung ist dabei leicht übertragbar auf unterschiedliche Systeme und Wechselwirkungen, ohne dass die Matrixelemente von Ein- und Zweiteilchenoperatoren jedes Mal in komplizierter Weise neu berechnet werden müssen. Außerdem lässt sich eine drastische Vereinfachung der Selbstenergien erzielen. Der zweite Punkt ermöglicht die Nutzung von Hoch- und Höchstleistungsrechnersystemen, deren Resourcen (in Computerleistung: hauptsächlich Speicher!) unabdingbar sind für (einige) der durchgeführten Rechnungen. Die Anwendbarkeit der entwickelten Methoden wird unter anderem für Helium, Beryllium, Wasserstoff und Lithiumhydrid gezeigt, wobei sich die Elektronen in einer Raumdimension bewegen und Elektron-Elektron-Korrelationen in der zweiten Bornschen Näherung behandelt werden. Nach einer sorgfältigen Diskussion der selbstkonsistenten Grundzustände beschäftigt sich die vorliegende Arbeit mit der Elektronendynamik sowohl im Linear Response Bereich als auch unter dem Einfluss starker externer Felder. Dabei werden die Resultate einerseits mit der Hartree-Fock-Näherung verglichen und andererseits der exakten Lösung der zeit(un)abhängigen Schrödinger-Gleichung gegenübergestellt, um die Güte der zweiten Bornschen Näherung zu untersuchen

Few-particle quantum dynamics–comparing nonequilibrium Green functions with the generalized Kadanoff–Baym ansatz to density operator theory

The nonequilibrium description of quantum systems requires, for more than two or three particles, the use of a reduced description to be numerically tractable. Two possible approaches are based on either reduced density matrices or nonequilibrium Green functions (NEGF). Both concepts are formulated in terms of hierarchies of coupled equations—the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced density operators and the Martin-Schwinger-hierarchy (MS) for the Green functions, respectively. In both cases, similar approximations are introduced to decouple the hierarchy, yet still many questions regarding the correspondence of both approaches remain open. Here we analyze this correspondence by studying the generalized Kadanoff–Baym ansatz (GKBA) that reduces the NEGF to a single-time theory. Starting from the BBGKY-hierarchy we present the approximations that are necessary to recover the GKBA result both, with Hartree-Fock propagators (HF-GKBA) and propagators in second Born approximation. To test the quality of the HF-GKBA, we study the dynamics of a 4-electron Hubbard nanocluster starting from a strong nonequilibrium initial state and compare to exact results and the Wang-Cassing approximation to the BBGKY hierarchy presented recently by Akbari et al. [1]

Quantum breathing mode of interacting particles in harmonic traps

The breathing mode – the uniform radial expansion and contraction of a system of interacting particles – is analyzed. Extending our previous work [Bauch et al 2009 Phys. Rev. B. 80 054515] we present a systematic analysis of the breathing mode for fermions with an inverse power law interaction potential w(r) ~ r−dwith d = 1,2,3 in the whole range of coupling parameters. The results thus cover the range from the ideal "gas" to the Wigner crystal-like state. In addition to exact results for two particles obtained from a solution of the time-dependent Schrödinger equation we present results for N = 4,6 from multiconfiguration time-dependent Hartree-Fock simulations

Nonequilibrium Green function approach to the pair distribution function of quantum many-body systems out of equilibrium

The pair distribution function (PDF) is a key quantity for the analysis of correlation effects of a quantum system both in equilibrium and far from equilibrium. We derive an expression for the PDF in terms of the single-particle Green functions—the solutions of the Keldysh/Kadanoff-Baym equations in the two-time plane—for a one- or two-component system. The result includes initial correlations and generalizes previous density matrix expressions from single-time quantum kinetic theory. Explicit expressions for the PDF are obtained in second Born approximation