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 an intuitive and graphical way 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 GW.

Markku Hyrkäs, Daniel Karlsson and Robert van Leeuwen, submitted for publication.


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 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.

Daniel Karlsson, Robert van Leeuwen, Enrico Perfetto, and Gianluca Stefanucci, Physical Review B 98 115148 (2018)


Non-equilibrium Green’s Functions for Coupled Fermion-Boson Systems

In this book section, we give a brief introduction to many-body perturbation theory for coupled fermion-boson systems using non-equilibrium Green’s functions. Using the language of modern many-body perturbation theory and the so-called contour-ordered correlators, a single consistent formalism arises which can be applied to a multitude of classes of systems.

Daniel Karlsson & Robert van Leeuwen, Handbook of Materials Modeling pp 1-29 (2018)



Time-dependent density-functional theory for strongly interacting electrons 

We consider an analytically solvable model of two interacting electrons that allows for the calculation of the exact exchange-correlation kernel of time-dependent density functional theory. This kernel, as well as the corresponding density response function, is studied in the limit of large repulsive interactions between the electrons and we give analytical results for these quantities as an asymptotic expansion in powers of the square root of the interaction strength. We find that in the strong interaction limit the three leading terms in the expansion of the kernel act instantaneously while memory terms only appear in the next orders. We further derive an alternative expansion for the kernel in the strong interaction limit on the basis of the theory developed previously [Phys. Chem. Chem. Phys. 18, 21092 (2016)] using the formalism of strictly correlated electrons in the adiabatic approximation. We find that the first two leading terms in this series, corresponding to the strictly correlated limit and its zero-point vibration correction, coincide with the two leading terms of the exact expansion. We finally analyze the spatial nonlocality of these terms and show when the adiabatic approximation breaks down. The ability to reproduce the exact kernel in the strong interaction limit indicates that the adiabatic strictly correlated electron formalism is useful for studying the density response and excitation properties of other systems with strong electronic interactions.

Luis Cort, Daniel Karlsson, Giovanna Lani, and Robert van Leeuwen, Physical Review A95, 042505 (2017)


Partial self-consistency and analyticity in many-body perturbation theory: Particle numberconservation 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 derivability for the self-energy 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 partially derivable. A special subclass of such approximations, which are gauge invariant, is obtained by dressing loops in the diagrammatic expansion of 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 whether commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial 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-semidefinite. The latter property can be ensured for a subset of partially -derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half diagrams. For the case in which the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure, and multivaluedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.

Daniel Karlsson and Robert van Leeuwen, Physical Review B94, 125124 (2016)

Vertex Corrections for Positive-Definite Spectral Functions of Simple Metals


We present a systematic study of vertex corrections in a homogeneous electron gas at metallic densities. The vertex diagrams are built using a recently proposed positive-definite diagrammatic expansion for the spectral function. The vertex function not only provides corrections to the well known plasmon and particle-hole scatterings, but also gives rise to new physical processes such as the generation of two plasmon excitations or the decay of the one-particle state into a two-particle–one-hole state. By an efficient Monte Carlo momentum integration we are able to show that the additional scattering channels are responsible for a reduction of the bandwidth, the appearance of a secondary plasmon satellite below the Fermi level, and a substantial redistribution of spectral weights. The feasibility of the approach for first-principles band-structure calculations is also discussed.

Yaroslav Pavlyukh, Anna-Maija Uimobnen, Gianluca Stefanucci, Robert van Leeuwen, Physical Review Letters 117, 206402, (2016)

Transient dynamics in correlated  quantum transport

We study quantum transport of interacting many-electron systems in the time-domain, with a focus on transient processes.dynamics.png

In the figure we show a chain of four atoms coupled to quasi two-dimensional electronic leads. The electrons are pushed through the four-atom chain by a bias voltage, where they interact with each other. These interactions are described by diagrammatic method in terms of Feynman diagrams. On the upper right hand side you can see the electronic density waves that appear in the electrodes in the course of time, whereas in the lower panels you can see how the currents develop in time for different voltages applied to the electrodes. The lower left figure is a mean field approach whereas the lower right one describes many-body electrons that go beyond mean field. As you can see, many-body interactions, considerably affect the properties of the molecular junction. On all the quantum transport work we have a wonderful collaboration with Gianluca Stefanucci of the University of Rome Tor Vergata, Italy. More information on the method and the applications can be found in our papers:

  • N. E. Dahlen and R. van Leeuwen, Solving the Kadanoff-Baym equations for inhomogeneous systems: Application to atoms and molecules, Phys. Rev. Lett. 98, 153004 (2007)
  • P. Myöhänen, A. Stan, G. Stefanucci, R. van Leeuwen, Conserving approximations in time-dependent quantum transport: Initial correlations and memory effects Europhys. Lett. 84, 67001 (2008)
  • P. Myöhänen, A. Stan, G. Stefanucci, R. van Leeuwen, Kadanoff-Baym approach to quantum transport through interaction nanoscale systems: From the transient to the steady-state regime Phys. Rev. B 80, 115107 (2009)
  • A.-M. Uimonen, E. Khosravi, A. Stan, G. Stefanucci, S. Kurth, R. van Leeuwen, Comparative study of many-body perturbation theory and time-dependent density functional theory in the out-of-equilibrium Anderson model Phys. Rev. B 84, 115103 (2011)


Coupled systems of electrons and phonons

Another application of Green's function methods is the study of electron-phonon coupled systems. Electrons in a crystal lattice or in a molecule do not experience a static field from the atomic nuclei but the nuclei move themselves as well. The nuclear motion is much slower than the electronic motion and therefore it is often a good approximation to use the Born-Oppenheimer approximation. However, there are many other situations in which nuclear motion leads to drastically new physics, the most well-known is probably the phenomenon of superconductivity. Also in many other situations the coupling between electrons and nuclear vibrations (also known as phonons) is of key importance. Electronic charges deform the local lattice or molecular structure and electrons can be dressed by lattice deformations to form a new quasi-particle known as the polaron that can considerably change the excitation properties of materials. In some organic molecules, such as polyacetylene, the polaron can have very special features and become a travelling soliton.

The molecular or lattice vibrations are quantized and can be theoretically described as bosonic field in which the number of bosons (excitations of the phonon modes) is not conserved (and hence the system does not Bose condense). The bosons interact with the electrons which, being fermions, are described by an anti-commuting field. This system of coupled fields can be studied by diagrammatic methods and the dynamics of coupled electron-phonon systems can be studied using time-propagation of the nonequilibrium Green's functions. In the figure we display how the properties of a single electron (described by the operators  and ) change when the coupling to a single phonon-mode is (described by the operators  and ) suddenly switched on. The energy of the electron (being  initially) is lowered by the dressing of phonons to form a new polaronic ground state. Also visible are phonon side bands that describe excited states of the polaron. On this topic we collaborate with Heiko Appel and Yang Peng of the Fritz-Haber Institute in Berlin, Germany. More information on our research in this topic is here:

  • N. Säkkinen and R. van Leeuwen, Electrons and Phonons with the Kadanoff-Baym formalism, poster Lausanne workshop on vibrational coupling, 2012


Nonlinear phenomena and bistability

The equations of motion for the particle and hole propagators in many-body theory are nonlinear integro-differential equations. The reason for the nonlinearity is that the Feynman diagrams are polynomials of the particle and hole propagators as a consequence of the various particle-particle scattering processes that they describe.

spectral.pngWhen we carry out re-summations to obtain dressed propagators we obtain consequently nonlinear self-consistent equations. It is well known that nonlinear differential equations can lead to important physical phenomena (e.g. solitons, chaotic systems). However, much less is known about nonlinear integro-differential equations. An important physical characteristic of the integral kernels that appear in such equations is that they carry memory. A simple physical picture that goes with this memory is that the way two particles interact or collide depends on how they collided earlier. When we completely neglect memory effects we are in the mean-field approximation. dens_five_all.pngWe can therefore ask whether the combination of nonlinearity and memory can lead to new non-equilibrium phenomena. We have investigated this for the case of a quantum transport system. We have been investigating this for the creation of steady-state currents in the out-of-equilibrium Anderson model (essentially a single interacting atom coupled to leads). We find that within the mean-field approximation we can push the system in multple steady states depending on how fast we switch on the bias voltage. In the figure we show the non-equilibrium spectral function in the time-dependent Hartree-Fock approxmation that illustrates how the atomic level switches in time to another energy state, while passing through a third unstable state. We found, however, that more sophisticated approach with memory kernels destroy the bistability. The project was carried out in collaboration with Gianluca Stefanucci of the University of Rome Tor Vergata, Italy, Stefan Kurth of the University of San Sebastian, Spain and Elham Khosravi and Hardy Gross of the Max-Planck Institute in Halle, Germany.

  • A.-M. Uimonen, E. Khosravi, G. Stefanucci, S. Kurth, R. van Leeuwen and E. K. U. Gross, Real-time switching between multiple steady-states in quantum transport, J. Phys. Conf. Ser. 220, 012018 (2010)
  • E. Khosravi, A. - M. Uimonen, A. Stan, G. Stefanucci, S. Kurth, R. van Leeuwen and E. K. U. Gross, Correlation effects in bistability at the nanoscale: steady state and beyond, Phys. Rev. B 85, 075103 (2012)



Theory of general initial states

The dynamics of a many-particle quantum state is an initial state problem. In experiment initial quantum states are often prepared such that they are eigenstates of specific operators. There are, however, also situations in which more general initial states are studied. This happens, for instance, in the relaxation dynamics of ultracold gases, response of nanoscale systems in nonequilibrium states and in optimal control theory. The solution of the initial state problem is obvious from the viewpoint of the time-dependent Schrödinger equation for which the evolution of the state is determined by time-propagation of a linear partial differential equation once we specific the initial state. The problem is that, in practice, we can not solve the time-dependent Schrödinger equation due to the large number of degrees of freedom in most many-body systems.

However, we have a number of methods based on reduced quantities, such as many-body Green's function theory. We can therefore as ourselves the question how to describe the initial state in this approach. If the initial state is a correlated ground state in thermodynamic equilibrium there is a special technique based on imaginary time propagation to calculate it (the so-called Matsubara technique). However, for more general initial states this is not possible. To incorporate such general initial states in the Green's function formalism one needs to specify initial n-body correlations by means of n-body density matrices. These density matrices appear as correlation blocks in the Feynman diagrams. The exact mathematical prescription for doing this was derived in our group in collaboration with Gianluca Stefanucci of the University of Rome Tor Vergata, Italy. The formalism will be applied to study systems in which the initial correlated state is hard to calculate in standard many-body theory. For more information see

  • R. van Leeuwen and G. Stefanucci, Wick theorem for general initial states, Phys. Rev. B 85, 115119 (2012)


Vertex corrections

The many-body approaches that we have been developing are based on expansion of the self-energy operator (which provides an effective field for added or removed particles) in powers of the particle and hole propagators (the Green's function) and the interaction. The self-energy can alternatively be expanded in terms of the Green's function G and the screened interaction W (the renormalized interaction between the electrons). This is of special importance in extended systems where bare interactions lead to infinities and the screened interactions presents a natural and physically interpretable quantity to expand in instead. In most of our calculations we have been using the lowest orders in this expansion (the so-called GW- approximation).sigmapskeleton.png

It is, however, known that a self-consistent solution of the GW equations in an extended system such as the electron gas can give unphysical spectral properties (washing out of plasmon structure). The natural step to go beyond GW and to possibly cure this problem would be to take the diagram to second order in the screened interaction (the so-called vertex diagram). Apart from being a higher order diagram, it also contains new physics. In the electron gas it would contain multiple electron-hole excitations and plasmon satellites that also have been observed in experiments on metals. Work on this project is in progress in collaboration with Yaroslav Pavlyukh of the University of Kaiserslautern, Germany.



Mathematical foundations of time-dependent density functional theory

The foundations of time-dependent density functional theory rest on two important assumptions

  • The uniqueness of a density-potential mapping for a given initial state
  • The existence of a Kohn-Sham system

Both assumptions can be proven under various mathematical conditions. The uniqueness was proven by Runge and Gross under the assumption of Taylor-expandability (and the subsequent assumption of mapping.pngconvergence of the Taylor series) of the external potential around the initial time. The existence of the Kohn-Sham system requires that a given density can be reproduced in a different system with different two-body interactions. This was proven under the same assumptions by our own research group. Recently we have been developing a new proof that widens the range of validity of the uniqueness and existence assumption. The proof is on a fixed point mapping between normed (Banach) spaces of potentials, and does not need the assumption of Taylor expandability. We have been further elucidating the structure of the (singular) Sturm-Liouville problem related to the spatial boundary conditions of the density-potential mapping. We further aim to elucidate for which class of potentials (appropriate Sobolev spaces) the uniqueness and existence theorems can be applied. Strong forces behind this project are our collaborators Michael Ruggenthaler and Markus Penz of the University of Innsbruck, Austria. For more information see:

  • R. van Leeuwen, Mapping from densities to potentials in time-dependent density-functional theory Phys. Rev. Lett. 82, 3863 (1999)
  • M. Penz and M. Ruggenthaler, Domains of time-dependent density-potential mappings J. Phys. A: Math. Theor. 44, 335208 (2011)
  • M. Ruggenthaler and R. van Leeuwen, Global fixed-point proof of time-dependent density-functional theory Europhys. Lett. 95, 13001 (2011)
  • M. Ruggenthaler, K. J. H. Giesbertz, M. Penz and R. van Leeuwen, Density-potential mappings in quantum dynamics Phys. Rev. A 85, 052504 (2012)


Numerical construction of density-potential mappings

The mathematical work on the foundations of time-dependent density functional theory has also provided a constructive scheme to construct the external potential that produces a given a time-dependent density. The method therefore solves the optimal control problem which potential produces a given time-dependent density profile for a given initial state. The method has the advantage that it can be applied to general initial states and systems with arbitrary interactions. Furthermore it is numerically stable even for large density changes. When applied to non-interacting systems it allows to construct the Kohn-Sham potential corresponding to a given density.

An example of our construction scheme is given in the figure. We consider two non-interacting particles on a one-dimensional ring. The initial state is a correlation two-body state which is not an eigenstate of the system. As a consequence this two-particle state develops in time with a strong time-dependence (see the figure in the left panel). We require, however, that the density is stationary and remains identical to its value at the initial time (see the inset panel in left hand side of the figure). To achieve this the external potential (which can not be stationary as the initial state is not an eigenstate) must have a very non-trivial time-dependence. We have therefore constructed a counter intuitive situation. Although the two-body correlations in the wave function have a strong time-dependence, this information is integrated out in the one-body density. The strong forces between this project are our collaborators Søren Nielsen of the University of Aarhus, Denmark and Michael Ruggenthaler of the University of Innsbruck, Austria. For more information see:

  • S. E. B. Nielsen, M. Ruggenthaler and R. van Leeuwen, Many-body quantum dynamics from the density, Europhys. Lett. 101, 33001 (2013)
  • M. Ruggenthaler, S. E. B. Nielsen, and R. van Leeuwen, Exact density functionals for correlated dynamics on a quantum ring, arXiv:1209.2949v2.