Partial self-consistency and analyticity in many-body perturbation theory: particle number conservation and a generalized sum rule

We consider a general class of approximations which guarantees the conservation of particle number in many-body perturbation theory. To do this we extend the concept of $\Phi$-derivability for the self-energy $\Sigma$ to a larger class of diagrammatic terms in which only some of the Green's function lines contain the fully dressed Green's function $G$. We call the corresponding approximations for $\Sigma$ partially $\Phi$-derivable. A special subclass of such approximations, which are gauge-invariant, is obtained by dressing loops in the diagrammatic expansion of $\Phi$ consistently with $G$. These approximations are number conserving but do not have to fulfill other conservation laws, such as the conservation of energy and momentum. From our formalism we can easily deduce if commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial $\Phi$-derivability plays an important role in the derivation of a generalized sum rule for the particle number, which reduces to the Luttinger-Ward theorem in the case of a homogeneous electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that the Green's function has certain complex analytic properties, which can be guaranteed if the spectral function is positive semi-definite.The latter property can be ensured for a subset of partially $\Phi$-derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half-diagrams. In case the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure and multi-valuedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.Comment: Minor changes, corrected typo

Solving the Kadanoff-Baym equations for inhomogenous systems: Application to atoms and molecules

We have implemented time-propagation of the non-equilibrium Green function for atoms and molecules, by solving the Kadanoff-Baym equations within a conserving self-energy approximation. We here demonstrate the usefulnes of time-propagation for calculating spectral functions and for describing the correlated electron dynamics in a non-perturbative electric field. We also demonstrate the use of time-propagation as a method for calculating charge-neutral excitation energies, equivalent to highly advanced solutions of the Bethe-Salpeter equation.Comment: 4 pages, 5 figure

Development of non-equilibrium Green's functions for use with full interaction in complex systems

We present an ongoing development of an existing code for calculating ground-state, steady-state, and transient properties of many-particle systems. The development involves the addition of the full four-index two electron integrals, which allows for the calculation of transport systems, as well as the extension to multi-level electronic systems, such as atomic and molecular systems and other applications. The necessary derivations are shown, along with some preliminary results and a summary of future plans for the code

Kadanoff-Baym approach to quantum transport through interacting nanoscale systems: From the transient to the steady-state regime

We propose a time-dependent many-body approach to study the short-time dynamics of correlated electrons in quantum transport through nanoscale systems contacted to metallic leads. This approach is based on the time-propagation of the Kadanoff-Baym equations for the nonequilibrium many-body Green's function of open and interacting systems out of equilibrium. An important feature of the method is that it takes full account of electronic correlations and embedding effects in the presence of time-dependent external fields, while at the same time satisfying the charge conservation law. The method further extends the Meir-Wingreen formula to the time domain for initially correlated states. We study the electron dynamics of a correlated quantum wire attached to two-dimensional leads exposed to a sudden switch-on of a bias voltage using conserving many-body approximations at Hartree-Fock, second Born and GW level. We obtain detailed results for the transient currents, dipole moments, spectral functions, charging times, and the many-body screening of the quantum wire as well as for the time-dependent density pattern in the leads, and we show how the time-dependence of these observables provides a wealth of information on the level structure of the quantum wire out of equilibrium. For moderate interaction strenghts the 2B and GW results are in excellent agreement at all times. We find that many-body effects beyond the Hartree-Fock approximation have a large effect on the qualitative behavior of the system and lead to a bias dependent gap closing and quasiparticle broadening, shortening of the transient times and washing out of the step features in the current-voltage curves.Comment: 16 pages, 14 figure

The Generalized Kadanoff-Baym Ansatz with Initial Correlations

Within the non-equilibrium 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 {\em 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 {\em 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

Implications of EU Enlargement for Agricultural Markets in the New Member States

The paper presents an analysis of the impact of the Common Agricultural Policy implementation on the agricultural markets of the eight new EU Member States. The study is based on the AGMEMOD (AGricultural MEmber states MODelling) national econometric models. Two scenarios are simulated for each country. The "Baseline" scenario assumes the implementation of the Single Area Payment Scheme until 2008 and the subsequent introduction of the Single Payment Scheme from 2009 onwards. Complementary national direct payments would remain in force until 2013. The second scenario assumes the full decoupling of direct payments from 2007 and the introduction of modulation from 2013 onwards in the 2004 enlargement new Member States (EU-8). The baseline scenario projections suggest that the introduction of direct payments would expand EU-8 aggregate production, mainly of oilseeds, grains, sheepmeat and cheese, while beef and veal production would also increase. Consumption of more expensive beef and veal meat would be substituted by poultry and pigmeat. Full decoupling of direct payments will have only a moderate impact on the balance of supply and use for crop and animal production.commodity markets, CAP reform, new Member States, econometric model, Marketing,

Conserving Approximations in Time-Dependent Density Functional Theory

In the present work we propose a theory for obtaining successively better approximations to the linear response functions of time-dependent density or current-density functional theory. The new technique is based on the variational approach to many-body perturbation theory (MBPT) as developed during the sixties and later expanded by us in the mid nineties. Due to this feature the resulting response functions obey a large number of conservation laws such as particle and momentum conservation and sum rules. The quality of the obtained results is governed by the physical processes built in through MBPT but also by the choice of variational expressions. We here present several conserving response functions of different sophistication to be used in the calculation of the optical response of solids and nano-scale systems.Comment: 11 pages, 4 figures, revised versio

Contour calculus for many-particle functions

In non-equilibrium many-body perturbation theory, Langreth rules are an efficient way to extract real-time equations from contour ones. However, the standard rules are not applicable in cases that do not reduce to simple convolutions and multiplications. We introduce a procedure for extracting real-time equations from general multi-argument contour functions with an arbitrary number of arguments. This is done for both the standard Keldysh contour, as well as the extended contour with a vertical track that allows for general initial states. This amounts to the generalization of the standard Langreth rules to much more general situations. These rules involve multi-argument retarded functions as key ingredients, for which we derive intuitive graphical rules. We apply our diagrammatic recipe to derive Langreth rules for the so-called double triangle structure and the general vertex function, relevant for the study of vertex corrections beyond the $GW$ approximation