Research interests in our research group
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.
Research topics
- Diagrammatic perturbation theory for non-equilibrium systems. We develop formalism and tools that can be used to study systems in which inter particle correlations and time-dependence play an important role. Recent developments include a new perturbation theory for positive spectral functions and number conservation in partially self-consistent theories.
- Fundamentals of time-dependent density-functional theory. We deepen the understanding of the mathematical foundations of density functional theory and develop new methods and functionals that are tested on solvable systems
- Interacting electron-boson systems. We study systems in which electron-boson interactions are important. Examples are molecules in optical cavities coupled to quantised photons and electrons coupled to vibrations in molecules and lattices.
- Many-body theory for strong-field pump-probe laser spectroscopy. We use first-principle non-equilibrium Green's function methods to study atoms and molecules in strong laser fields and study time-dependent ionisation.
Collaboration
- Gianluca Stefanucci (Diagrammatic many-body theory)
- Yaroslav Pavlyukh (Diagrammatic many-body theory)
- Michael Ruggenthaler (Time-dependent density-functional theory)
- Klaas Giesbertz (Density functionals and density matrices)
- Giovanna Lani (Time-dependent density-functional theory)
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 10^{23}. 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.
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)
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)