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arXiv:0905.2061 · Physical Review Letters

Successes and Failures of Kadanoff-Baym Dynamics in Hubbard Nanoclusters

M. Puig von Friesen, Claudio Verdozzi, C.‐O. Almbladh

Published Oct 23, 2009• 122 citations
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Abstract

We study the nonequilibrium dynamics of small, strongly correlated clusters, described by a Hubbard Hamiltonian, by propagating in time the Kadanoff-Baym equations within the Hartree-Fock, second Born, GW, and T-matrix approximations. We compare the results to exact numerical solutions. We find that the time-dependent T matrix is overall superior to the other approximations, and is in good agreement with the exact results in the low-density regime. In the long time limit, the many-body approximations attain an unphysical steady state which we attribute to the implicit inclusion of infinite-order diagrams in a few-body system.

DOI
10.1103/physrevlett.103.176404

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We present in detail a method we recently introduced (PRL. 103, 176404 (2009)) to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym Equations (KBE) within Many-Body Perturbation Theory (MBPT). The main property we analyze is the time-dependent density. We also study is the exchange-correlation potential of TDDFT, obtained via reverse engineering from the time-dependent density. Our systems consist of small, strongly correlated clusters, described by a Hubbard Hamiltonian within the Hartree-Fock, second Born, GW and T-matrix approximations. We compare the results from the KBE dynamics to those from exact numerical solutions. The outcome of our comparisons is that, among the many-body schemes considered, the T-matrix approximation is overall superior at all electron densities. Such comparisons permit a general assessment of the whole idea of applying MBPT, in the KBE sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the KBE develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated (finite) systems, where it is purely artificial, as well as in clusters contacted to (infinite) macroscopic leads. To illustrate this point we present selected results for a system coupled to contacts within the T-matrix and second Born approximation. The extensive characterization we performed indicates that this behavior is present whenever approximate self energies, based upon infinite partial summations, are used. A second important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. This may also true for clusters connected to leads.

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We study the nonequilibrium dynamics of small, strongly correlated clusters, described by a Hubbard Hamiltonian, by propagating in time the Kadanoff-Baym equations within the Hartree-Fock, second Born, GW, and T-matrix approximations. We compare the results to exact numerical solutions. We find that the time-dependent T matrix is overall superior to the other approximations, and is in good agreement with the exact results in the low-density regime. In the long time limit, the many-body approximations attain an unphysical steady state which we attribute to the implicit inclusion of infinite-order diagrams in a few-body system.
We present in detail a method we recently introduced (PRL. 103, 176404 (2009)) to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym Equations (KBE) within Many-Body Perturbation Theory (MBPT). The main property we analyze is the time-dependent density. We also study is the exchange-correlation potential of TDDFT, obtained via reverse engineering from the time-dependent density. Our systems consist of small, strongly correlated clusters, described by a Hubbard Hamiltonian within the Hartree-Fock, second Born, GW and T-matrix approximations. We compare the results from the KBE dynamics to those from exact numerical solutions. The outcome of our comparisons is that, among the many-body schemes considered, the T-matrix approximation is overall superior at all electron densities. Such comparisons permit a general assessment of the whole idea of applying MBPT, in the KBE sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the KBE develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated (finite) systems, where it is purely artificial, as well as in clusters contacted to (infinite) macroscopic leads. To illustrate this point we present selected results for a system coupled to contacts within the T-matrix and second Born approximation. The extensive characterization we performed indicates that this behavior is present whenever approximate self energies, based upon infinite partial summations, are used. A second important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. This may also true for clusters connected to leads.
We present in detail a method we recently introduced [Phys. Rev. Lett. 103, 176404 (2009)] to describe finite systems in and out of equilibrium, where the evolution in time is performed via the Kadanoff-Baym equations within many-body perturbation theory. Our systems consist of small, strongly correlated clusters, described by a Hubbard Hamiltonian within the Hartree-Fock, second Born, GWGWGW, and TTT-matrix approximations. We compare the results from the Kadanoff-Baym dynamics to those from exact numerical solutions. The outcome of our comparisons is that, among the many-body schemes considered, the TTT-matrix approximation is superior at low electron densities while none of the tested approximations stands out at half filling. Such comparisons permit a general assessment of the whole idea of applying many-body perturbation theory, in the Kadanoff-Baym sense, to finite systems. A striking outcome of our analysis is that when the system evolves under a strong external field, the Kadanoff-Baym equations develop a steady-state solution as a consequence of a correlation-induced damping. This damping is present both in isolated (finite) systems, where it is purely artificial, as well as in clusters contacted to (infinite) macroscopic leads. The extensive numerical characterization we performed indicates that this behavior is present whenever approximate self-energies, which include correlation effects, are used. Another important result is that, for isolated clusters, the steady state reached is not unique but depends on how one switches on the external field. When the clusters are coupled to macroscopic leads, one may reach multiple quasisteady states with arbitrarily long lifetimes.
We perform a comparative study of exact and approximate time-evolved densities in small Hubbard chains. The approximate densities are obtained via many-body perturbation theory (Hartree-Fock, 2nd Born, GW and T-matrix approximations) within the framework of the time-dependent Kadanoff-Baym equations. Benchmarking approximate results against exact ones allows us to address two rather fundamental issues in the non equilibrium dynamics of strongly correlated systems. I) A characterisation of the performance of several standard MBAs in the non-equilibrium regime. Having a definite notion of how good a specific MBA can be is highly relevant to its application to cases (typically, infinite systems) where exact solutions are not available. Our results show that the T-matrix approximation is overall superior to the other MBAs, at all electron densities. II) A scrutiny of the whole idea of Many Body Perturbation Theory in the Kadanoff-Baym sense, when applied to finite systems. The surprising outcome of our study is that during the time evolution, the KBE develop an unphysical steady state solution. This is a genuinely novel feature of the time-dependent KBE, i.e. is not inherited from possible limitations/approximations in the calculation of the initial state. Our extensive numerical characterisation gives robust evidence that the problem occurs in general, whenever MBPT is applied to finite systems, and approximate self energies based upon infinite partial summations are used. We also offer some more conceptual and general consideration on the dependence of this behaviour on the number of particles and system size. This is followed by our conclusions and glimpses of future work.
The Hubbard model is a prototype for strongly correlated many-particle systems, including electrons in condensed matter and molecules, as well as for fermions or bosons in optical lattices. While the equilibrium properties of these systems have been studied in detail, the nonequilibrium dynamics following a strong nonperturbative excitation only recently came into the focus of experiments and theory. It is of particular interest how the dynamics depend on the coupling strength and on the particle number and whether there exist universal features in the time evolution. Here, we present results for the dynamics of finite Hubbard clusters based on a self-consistent nonequilibrium Green functions (NEGF) approach invoking the generalized Kadanoff-Baym ansatz (GKBA). We discuss the conserving properties of the GKBA with Hartree-Fock propagators in detail and present a generalized form of the energy conservation criterion of Baym and Kadanoff for NEGF. Furthermore, we demonstrate that the HF-GKBA cures some artifacts of prior two-time NEGF simulations. Besides, this approach substantially speeds up the numerical calculations and thus presents the capability to study comparatively large systems and to extend the analysis to long times allowing for an accurate computation of the excitation spectrum via time propagation. Our data obtained within the second Born approximation compare favorably with exact diagonalization results (available for up to 13 particles) and are expected to have predictive capability for substantially larger systems in the weak-coupling limit.
This dissertation investigates the possibilities and limitations of time-dependent many-body perturbation theory by studying small Hubbard clusters for which the exact solution is available. The first part of the thesis is comprised of a short introduction to the concepts and methodologies used. The second part consists of an review of the main findings of the thesis and a short summary of four original papers. In paper I we study the dynamics of short Hubbard chains within many-body perturbation theory and compare to the exact solution. The main outcomes are that the T-matrix approximation works well in the low filling regime and that all approximation which involve correlation effects develop an unphysical steady state. In paper II we present the methodology used in paper I. We find that the correlation-induced damping appears even in the presence of leads and that there are multiple steady and quasi steady states. In paper III we make a review of the status of time dependent density functional theory for lattice models. In particular we find that a non-perturbative adiabatic local density approximation describes strong correlations well while many-body perturbation theory accounts better for non-adiabatic effects. In paper IV we propose a method to obtain the double occupancy from the Kadanoff-Baym equations. We show that the positiveness condition may be violated in the GW or the second Born approximation but fulfiled in the T-matrix approximation. We apply this method to obtain the local entanglement entropy.
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].