# Quantum Many-Body Theory

The main focus of this group is to study quantum non-equilibrium systems and to develop the underlying theoretical tools and methods. The main approaches to non-equilibrium physics are Time-Dependent Density Functional Theory (TDDFT) and Nonequilibrium Green's Function theory (NEGF). As an application the Green's function method is applied to describe quantum transport through molecular devices.

 Current group members Alumni Robert van Leeuwen (group leader) Markku Hyrkäs Luis Cort Daniel Karlsson

### Material

What is quantum many-body theory?

All objects we see around us everyday are made up of electrons and atomic nuclei. Questions about the nature of these objects, such as their color, the question whether the object is a solid, liquid or a gas, or whether it may be superconducting or ferromagnetic are ultimately questions about the quantum mechanical behavior of systems made up of electrons and atomic nuclei. These systems may not be in a stationary state. They may, for instance, be excited by lasers or carry electronic currents and therefore be in a dynamical state. To describe such phenomena we need a dynamical description of quantum many-particle systems. In principle we know the quantum mechanical laws that describe the dynamical evolution of these systems. To predict the properties of such systems we simply need to solve the time-dependent Schrödinger equation. The problem is that this is not possible in practice. First of all, in the range from atoms and molecules to solids we are dealing with systems with an electron number (and the same for nuclei) ranging from 1 to 1023. Which means that we are dealing with quantum mechanical wave functions of an enormous number of variables. Secondly, all these particles are interacting with Coulombic forces such that the motion of the particles are not independent and therefore equations can not be simplified. We are therefore faced with the question how to theoretically predict properties of such systems. This problem is commonly known as the many-body problem. It is a central problem in theoretical physics and an enormous amount of work has been done to attack this problem. One key observation is that most experimental properties of many-particle systems involve one and two-body observables such as densities and currents, polarizabilities, spin quantum numbers, pair correlation functions etc. This suggests that it may be possible to describe the properties of many-body systems in terms of reduced quantities, i.e. quantities that are obtained by integrating out all but a few of the variables of the many-body wave function. This idea has turned out to be very fruitful and has led to various theoretical approaches to attack the many-body problem. These approaches are nonequilibrium Green's function theory, density functional theory and density matrix theory. In our research group all these three approaches are developed and applied to the study of many-particle systems.

Lectures on Many-Body Theory (Benasque school on TDDFT 2014)

Nonequilibrium Green's function theory

Nonequilibrium Green's functions is a general perturbative approach to calculate time-dependent observables of quantum many-body systems. The theory is based on expansion of these observables in powers of the two-body interactions, while external time-dependent fields are treated exactly. Often it turns out that order by order perturbation theory is insufficient. Therefore perturbative series are summed to infinite order. A key role in carrying out such resummations is played by the self-energy operator and the Dyson equation. In nonequilbrium theory the Dyson equation is translated into a set of integro-differential equations, known as the Kadanoff-Baym equations, that need to be solved by time-propagation. A pedagogical introduction to the whole theory for student with a background in standard quantum mechanics is given in

• G. Stefanucci and R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems, Cambridge University Press, Cambridge, UK (2013)

Dynamics of open quantum systems

An important application of nonequilibrium Green's theory is the study open quantum systems. An important class of open quantum systems is that of a molecule coupled to a macroscopic electronic system such as a metallic surface, or coupled between two macroscopic metallic electrodes that can act as a reservoir for electrons. The latter situation is heavily studied in the field of molecular electronics in which one studies electron transport through single molecules upon applications of bias voltages to the macroscopic electrodes. To theoretically describe the time-dependent processes that happen in the molecule is a very challenging task. One deals with many-body interactions in quantum systems which are far from equilibrium and in which the number of electrons is not preserved. We study such systems using nonequilibrium Green's functions, by means of time-propagation schemes. The Green's functions are an ideal tool such they describe the addition and removal of electrons to the molecule. This is a very natural tool to use in a system where the number of electrons fluctuate.

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 $\hat{b}$ and $\hat{b}^\dagger$) change when the coupling to a single phonon-mode is (described by the operators $\hat{a}$ and $\hat{a}^\dagger$) suddenly switched on. The energy of the electron (being $\epsilon_0=0$ 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.

When 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. We 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. From 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).

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 Halle, Germany.

Exact diagonalization

To test our many-body approaches or to get additional insight into the structure of the many-body states we also carry out exact diagonalizations of simplified Hamiltonians for which the Hilbert space is finite (but quite large). One often maps the Hamiltonian to a lattice model, in which every atomic site can occupy at most two electrons (generalizations to more orbitals per site is also possible). If we deal with N atomic sites over which we have to distribute $N_{\uparrow}$ electrons with $S_z$ quantum number +1/2 and $N_{\downarrow}$ electrons with quantum number -1/2 then the size of the Hilbert space is

which is a number that grows very rapidly with the number of electrons which again illustrates the many-body problem. In the figure we show a simple illustration of a simple lattice with one particular configuration of 6 electrons on 11 sites which is one basis state in the 27225-dimensional space of states.

By exact diagonalization we have full access to all many-body states. This can be used to calculate any many-body observable or correlation function. Moreover, we can construct solutions of the time-dependent Schrödinger equation to do dynamics. of course, this is only solvable for small lattices. For instance, if we put 5 up and 5 down electrons on a 20 site lattice (a common half-filling situation) our Hilbert space already becomes 33802596-dimensional. There is a clear need for methods to study larger many-body systems.

Density functional theory

In density functional theory the basic variable is the density which gives the probability to find a particle at a given space-time point. It is a simple quantity of one space-time variable, instead of the Green's functions which depends on two space-time points. The basic underlying idea of density functional theory is that the external potential of a many-body system, for given two-body interactions, is uniquely determined by its density (and for time-dependent systems also its initial state). This implies that the many-body states, and therefore all observables, are functionals of the density. Density functional theory is usually applied by defining an auxiliary system of noninteracting particles, known as the Kohn-Sham system, having the same density as the interacting system that it models. The effective one-particle potential in this auxiliary system contains all information on the many-body system and is a functional of the density. By solving the Kohn-Sham equations the exact density of the interacting can be determined. The one-body structure of the Kohn-Sham equations make the theory very attractive from a computational point of view. This fact, as well as the possibility of having an exact one-particle picture, to a large extent has accounted for the popularity of the theory. It can therefore be applied to large systems with often good accuracy. There are, however, notable failures as well and the development of more accurate and relatively easy applicable density functionals to describe electron correlations remains a large theoretical challenge. Our own research has mainly been focussing on the time-dependent density functional and in particular its mathematical foundations, but we have also been investigating nonlocal density functionals in general. A very pedagogical introduction to time-dependent density functional theory and many of its applications is given in

• Carsten Ullrich, Time-dependent density-functional theory, Oxford University Press, Oxford (2012)

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 convergence 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.

Development of approximate density functionals

The formalism of density functional theory would be of little use if there were no useful applications. The wide-spread use of density functional theory that for many applications (mostly to molecules, solids and surfaces) useful predictions can be made. Most of these predictions are made (at least in the ground state theory) using the so-called local density approximation and generalized gradient functionals. However, in important cases these approximate density functionals can fail miserably. These cases usually involve situations in which the many-body wave function is described by a linear combinations of states with strong mixing (when expanded in one-particle states). Examples are found in open-shell systems with close degeneracies and in the dissociation of molecules which are obviously of large interest to chemists. These situations are more naturally described in wave function or many-body theories but unfortunately many systems of interest are too large to use such methods in practice. It is therefore worthwhile also to explore whether we can find density functional that can also describe such strongly correlated systems. One can expect that simple local and semi-local approximations based on the local densities and its gradients must be abandoned and one must search for truly nonlocal approximations. Important insight into density functionals is provided by the pair-correlation function or the exchange-correlation hole. These functions describe the two-body correlations between electrons and from wave function calculations a lot of insight has been gained in the nature of these functions.

We are currently developing an approximate density functional based on a non-local model of the exchange-correlation hole but current results are very encouraging. The equations to be solved are nonlinear integral equations which probe the density everywhere in pace, leading to a very nonlocal density functional. As an example of the functions that we model we show in the figure the exchange hole, the correlation hole and its sum for a dissociating hydrogen molecule at bond distance of 5 bohr. The work is a collaboration between Klaas Giesbertz of the Free University of Amsterdam, The Netherlands and Ulf von Barth of the University of Lund, Sweden. For more information see:

• K. J. H. Giesbertz, R. van Leeuwen and U. von Barth, Towards nonlocal density functionals by explicit modeling of the exchange-correlation hole in inhomogeneous systems, Phys. Rev. A 87, 022514 (2013)
• U. von Barth, Basic Density-Functional Theory- An Overview, Physica Scripta T 109, 9 (2004)

Connections to many-body perturbation theory

Density functional theory is an exact formalism to calculate the properties of many-body systems. However, the way the density functionals are defined is very implicit. This implicitness makes it hard to find good approximate density functionals. This is even more so in time-dependent density functional theory where there are non-localities in space and time which are strongly intertwined. The structure of time-dependent density functionals can be greatly clarified by making connections to many-body perturbation theory. For instance, the memory structure of of density functionals is a direct consequence of electronic interactions. In diagrammatic perturbation theory these interactions carry memory as the various particle-particle collision happen at vertices in Feynman diagrams that need to be integrated over internal times. For instance, in the figure below it is shown how the exchange-correlation kernel of time-dependent density functional theory, can be expressed in diagrams at a GW level of many-body perturbation theory.

Also the initial state dependence can be elucidated in the same way. The initial correlations appear as correlation blocks in the diagrams that also carry memory. A further advantage is that in many-body theory we know how to select diagrams in such a way that basic conservation laws (energy, momentum etc.) are obeyed. In time-dependent density-functional theory these conditions appear as constraints on the exchange-correlation potential. Approximations satisfying these constraint can be directly derived from many-body theory. The main virtue of the connection to many-body Green's functions is the insight. How to turn such functionals into sufficiently approximate schemes with more favorable computational cost than many-body perturbation theory itself is very much an open problem. However, for the calculation of ground state energies, the use of so-called Luttinger-Ward functionals in connection with density functional theory is very useful and computational cheap method. For more information see:

• R. van Leeuwen, The Sham-Schlüter equation in time-dependent density-functional theory, Phys. Rev. Lett. 76, 3610 (1996)
• R. van Leeuwen, N. E. Dahlen, G. Stefanucci, C.-O. Almbladh and U. von Barth, Introduction to the Keldysh formalism, in Time-Dependent Desnity Functional Theory, eds. M. Marques et al., Lecture Notes in Physics 706, Springer (2006), (http://arxiv.org/abs/cond-mat/0506130v1)
• U. von Barth, N. E. Dahlen, R. van Leeuwen and G. Stefanucci, Conserving approximations in time-dependent density-functional theory, Phys. Rev. B72, 235109 (2005)
• N. E. Dahlen, R. van Leeuwen and U. von Barth, Variational functionals of the Green function and of the density tested on molecules, Phys. Rev. A 73, 012511 (2006)

Density matrix theory

The hierarchy of coupled density matrices

The BBGKY hierarchy for the density matrices is a set of equations that couples the equations of motion of the n-particle density matrix to that of the n+1-particle density matrix. The hierarchy is, in fact, a special case of the hierarchy equations for the n-particle Green's function (the so-called Martin-Schwinger hierarchy).

The advantage of the BBGKY hierarchy is that the equations are dependent on a single time-variable as apposed to that of the many-particle Green's function. The disadvantage is that, opposed to the case of many-body Green's function theory, there is no systematic perturbation theory for a higher order density matrix in terms of lower order ones. This means that the decoupling of the BBGKY hierarchy must be based on less well-founded assumptions (usually analogies with expansions for the Green's function are used). Moreover, the generation of the properly correlated initial state, is another problem that has no simple solution. We have tried a number of such decoupling schemes that express the 3-particle density matrix in terms of 1- and 2-particle density matrix and tested this for some time-dependent states of the Hubbard model. We found, however, that this must be done in a specific way to satisfy the energy conservation law. We further found that the solutions lead to unphysical divergencies in time, closely related to the fact that we solve nonlinear differential equations. It seems that we need to impose some extra physical constraints to our equations. How this needs to be done is an open question at the moment. For more information on our efforts in this direction see

• A. Akbari, M. J. Hashemi, A. Rubio, R. M. Nieminen and R. van Leeuwen Challenges in truncating the hierarchy of time-dependent reduced density matrices equations Phys. Rev. B 85, 235121, (2012)
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