STATISTICAL PHYSICS OF COMPLEX SYSTEMS
Here, you find an overview of our research interests. We discuss
some projects and also list some current collaborations. There are
several recent articles which you can download.
Our complete list of publications can be found
here.
Quantum Chaos and Random Matrix Models
Random matrices provide powerful models for a rich variety of complex
systems. Here is a brief
overview.
A good example is the atomic nucleus shown to the right. It consists
of many nucleons, the protons (blue) and the neutrons (red). They all
interact with each other and move in a very complicated way. Large
parts of the quantum mechanical excitation spectrum have statistical
features which are well described by assuming that the interaction
matrix elements can be replaced by random numbers. Hence, the
Hamiltonian can be viewed as a random matrix. One often refers to
this as Quantum Chaos in nuclei.
In several different branches of physics, we study chaotically
coupled systems or, equivalently, systems with symmetry breaking. The
coupling or the symmetry breaking has a strong influence on the
spectral statistics. In a recent work, we look at the time evolution of a
certain
energy localization effect which has been observed
in coupled elastodynamical systems.
Recently, we carried out detailed studies of the doorway mechanism in the
framework of Random Matrix Theory. Doorways are distinct states in
complex quantum systems which couple to a background of other states
which often can be modeled by random matrices. Important
conclusions about various systems can be drawn once the statistical
implications of the doorway mechanism have been understood.
The
superscars in the barrier billiard provide an
ideal model scenario for an in-depth investigation.
In this context, we also calculated the
distribution of the coupling coefficients to the doorway.
The doorway mechanism is closely related to an important question of
quantum information, namely how a prepared state is influenced by
the mixing to its environment. We find a striking
recovery and saturation
of the survival probability.
collaborators:
Johan
Grönqvist (PhD-student, now PhD),
Dr. Heiner Kohler
(now Madrid),
Professor Hans-Jürgen Sommers
external collaborators:
Professor
Sven Åberg at University of Lund,
Professor
Achim Richter and his group at Technical University of Darmstadt,
Professor
Hans-Jürgen Stöckmann at University of Marburg.
Many-Body Systems and Quantum Chaos

As discussed above, nuclei very often show statistical features
which are well described by random matrices. However, that is not the full story
yet, because at some excitation energies, the nucleons move
coherently in phase space. This leads, for example, to vibrations and rotations of
the nucleus as a whole. Those collective excitations are not quantum
chaotic, they rather tend to behave regularly.
Proper Random Matrix Models can be used to distinguish
collective from quantum chaotic motion, to some extent even purely
from the measured data. We analyzed experimental data of
magnetic scissors mode excitations and of
electric pygmy dipole excitations.
This also means that the strong arguments recently put forward
for single-particle systems in favor of the Bohigas-Giannoni-Schmit
conjecture cannot be carried over in a straightforward way to many-body
systems. (The Bohigas-Giannoni-Schmit conjecture states that
the spectral fluctuations of a quantum system are described by random matrices
if the corresponding classical system is chaotic in the usual sense
of classical mechanics.) We devote considerable efforts to the understanding of
the interplay of incoherent single-particle and collective motion
as well as its impact on the level statistics in many-body systems.
We began with investigating
spreading and its semiclassical interpretation
in a model system in which we always have control over the individual particle
dynamics. We derive, under rather general circumstances, that the collective
observables of the quantum system relate to the classical dynamics.
collaborators: Johannes Freese (Diplom-student),
Dr. Boris Gutkin (junior research group leader),
Jens Hämmerling (PhD-student)
external collaborators:
Professor
Achim Richter and his group at Technical University of Darmstadt
Quantum Chaos Approach to QCD
Quantum Chromodynamics (QCD) is the theory of the strong interaction.
The quarks (the colored spheres) interact by exchanging gluons which
are described by gauge fields (the green clouds). Unfortunately, QCD
is so involved that analytical calculations are only feasible after
drastic approximations. Thus, one resorts to demanding numerical
simulations, referred to as lattice gauge calculations. We contributed
to the statistical analysis of those lattice data which show that QCD
has much in common with disordered mesoscopic systems in condensed
matter physics. But QCD is even more complicated, in particular due to
chirality and chiral symmetry breaking. We setup an effective
stochastic field theory. Presently, we
investigate the motion of the quarks in individual gauge field
configurations which matches the way how lattice gauge calculations
are done. As a first step, we identified a
hierarchy of semiclassical limits in QCD.
collaborator:
Dr. Stefan
Keppeler (Emmy Noether fellow, now at University of Tübingen)
external collaborator:
Professor Tilo Wettig at University of
Regensburg.
Supersymmetry and Supergroups
Supersymmetry became a prominent tool in the Theory of Random Matrices
and Disordered Systems. We developed the Graded Eigenvalue Method as a
new and exact technique to solve Random Matrix Models. In the course
of doing so we were led to address the general problem of harmonic
analysis on superspaces and, in this context, the theory of
supergroups. Here is an
introduction to the main ideas. We
showed that certain matrix Bessel functions (in mathematics referred
to as Gelfand's spherical functions) are kernels of diffusion
equations which drive systems with arbitrary spectral correlations
into the chaotic regime. We derived various new and explicit results
beyond the unitary case for these diffusion kernels in
ordinary and
super spaces. As a side result, we also
proved a
supersymmetric generalization of
Harish-Chandra's famous integral formula.
There is a close relation of these issues to Calogero-Sutherland
models. We constructed a large class of those models in superspace.
In a most natural way, models for
two kinds of interacting
particles result which even have a physical interpretation.
For a long time, one thought that supersymmetry is restricted to Gaussian
probability densities. We showed
arbitrary probability densities can be treated with supersymmetry.
Later on, this was also studied in the framework of superbosonization. We showed
the
equivalence.
collaborators:
Dr. Mario Kieburg (PhD-student, now PhD),
Dr. Heiner Kohler
(now Madrid),
Professor Hans-Jürgen Sommers
Applications of Supersymmetry
Supersymmetry is nowadays indispensable for Random Matrix Theory,
because it yields an exact representation of many Random Matrix Models
in terms of supermatrix models in such a way that the number of
degrees of freedom is drastically reduced. We use this in many applications.
Some examples may illustrate that.
Consider a Hamiltonian that depends on some parameter. The correlations in this parameter
are known to have universal features. Another observable is fidelity. It measures the overlap
between states, evolved with a perturbed and an unperturbed Hamiltonian, depending on the
perturbation parameter. Hence, in some way, parametric correlations and fidelity must be
related, but it was unclear how. Supersymmetry helped us to discover surprising relations
between
parametric level correlations and fidelity decay.
When analyzing time series, for example of the electrodes attached to different places on the scalp
in electroencephalography (EEG), the correlations coefficients between two time series provide
important information about the system. The correlation coefficients are ordered in the correlation
matrix. The Wishart model is a benchmark for statistical properties of such correlation matrices.
It correctly incorporates the empirical eigenvalues, but is otherwise based on Gaussian
distributed random time series. Formally, the Wishart model coincides with chiral Random Matrix
Theory, if the empirical eigenvalues are all equal. An important observable is the density of
eigenvalues of the correlation matrix which parametrically depends on the empirical
eigenvalues. For complex correlation matrices, this eigenvalue density was known exactly, but in the
real case, a deep mathematical reason made it impossible for a long time to derive exact
closed form results. Using supersymmetry, we fully solved this problem and gave exact formulas for the
eigenvalue density
of real Wishart correlation matrices.
Structural insights are often also useful in applications. We discovered an
unexpected connection between the
building blocks of the correlation functions. Furthermore, we found
supersymmetric structures without actually mapping to superspace.
More precisely, we identified superspace Jacobians in the integral representations
of correlation functions in ordinary space. This deep structural insight
yields a powerful method to cast correlation functions into a
determinant or
Pfaffian form, depending on the underlying symmetric spaces.
collaborators:
Johan
Grönqvist (PhD-student, now PhD),
Dr. Mario Kieburg (PhD-student, now PhD),
Dr. Heiner Kohler
(now Madrid),
Christian Recher (Diplom-student, now PhD-student with Dr. Heiner Kohler)
external collaborators:
Dr. Francois Leyvraz and
Dr. Thomas Seligman and their groups at UNAM, Cuernavaca (Mexico),
Dr. Igor Smolyarenko at Brunel University, Uxbridge (England)
Elasticity, Wave Chaos and Ray Limit
Classical wave phenomena are ubiquitous in technical applications.
Quantum Chaos is the statistical theory of quantum waves. What can we
learn from Quantum Chaos for classical wave phenomena? - We addressed
this question by studying elastomechanical systems in a series of
investigations over the years. This is non-trivial due to the presence
of modes (pressure and shear) and due to the complicated boundary
conditions. Among other things, we presented the first purely
experimental and statistically highly significant study of
parametric correlations. More recently, we
showed that the elastic displacement field behaves statistically just
like a quantum wave function. However, there is a
beating phenomenon between the different modes.
We now apply "semiclassical" ideas, that is, we study the ray limit of
elastodynamics in which the wavelength is small compared to all
system geometries. A first step was a
Weyl approximation to the level density of the
quartz sphere shown to the right. This is
a monocrystal of grapefruit size whose spectra have been measured with
fantastic precision.
More ambitiously, we now work on understanding of
elastodynamical spectra by relating fundamental features of those
systems to geometric quantities such as periodic orbits.
In particular, we became interested in shells because of their
relevance for all kinds of applications, ranging from auto bodies to rockets.
We considered a family of shells of revolution, with the disk and the hemisphere as limiting cases. We
managed to explain a
striking clustering effect in the spectra using periodic orbits.
collaborators:
Dr. Niels Søndergaard (postdoc, now researcher in industry)
external collaborators:
Professor Clive Ellegaard, Mikkel Avlund at the Niels
Bohr Institute, Copenhagen (Denmark),
Dr. Mark Oxborrow at the National
Physical Laboratory, Teddington (England).
Micro- and Nanomechanics
Little is known about the mechanical properties of systems on the
micrometer and nanometer scale. An example is the cantilevers shown to
the left. Nonlinear or chaos-related effects are needed to gain deeper
insight into those systems. We wish to apply our knowledge of chaos
theory and elastodynamics to the micrometer and nanometer scale. In
particular, we are interested in micro-electro-mechanical systems
(MEMS) and nano-electro-mechanical systems (NEMS) which are nowadays
so important in engineering. We used Euler's theory for rods to understand
mechanical properties of cantilevers as
shown in the picture.
The continuum theory of elasticity is bound to fail at those scales
where the atomistic structure of matter becomes relevant. It is
important to figure out on which scales this happens. We studied
strain in semiconductor core--shell nanowires
in an atomistic and in a continuum model. We found a remarkable
robustness of the continuum theory.
collaborators:
Johan
Grönqvist (PhD-student, now PhD),
Dr. Niels Søndergaard (postdoc, now researcher in industry)
external collaborators:
Professor Lars Montelius and
Professor Hongqi Xu
and their groups at the
Lund Nano Lab in Lund (Sweden).
Econophysics
What is theoretical physics? - Our definition is the following:
Theoretical physics is construction and analysis of mathematical
models for the quantitative description of reproducible
experiments. These models ought to be as compatible as possible with
each other to yield an ever more unifying picture for the measurable
features of our world. - There is no reason to
exclude biology, sociology or economics from the list of interesting
research topics. On the contrary, we think that we physicists would
make a big strategic mistake by confining ourselves to the more
established topics and by leaving, for example, the quickly expanding
field of mathematical modeling in economics to the mathematicians. The
physicists' expertise, in particular the data- and experiment-oriented
approach has a truly high potential in this field.
The time series of stocks are correlated, because the performance of
the corresponding companies is mutually connected. The measurement of
those financial correlations is very important for risk management. For
various reasons, the empirically obtained correlations are dressed
with noise. We developed methods to estimate and remove this noise, to
begin with in a
random matrix approach and then by
inventing the
power mapping as an alternative and supplement.
These noise reduction techniques have direct
applications in portfolio optimization and can greatly reduce risk in stock portfolios.
An interesting feature of financial correlation is their behavior on short time scales. For relative price changes (or
returns) on very short time intervals, the influence of
asynchrony in trading and the
finite tick size become increasingly important. They are the main statistical causes for the so-called Epps-effect, i.e., for decreasing correlations on short return intervals.
In addition to market risk we also study
credit risk. Since an obligor can go
bankrupt or default in another way, credits are risky for the banks
that issue them. It turns out that the distributions of the losses are
highly asymmetric and have long tails. A better quantitative
understanding by means of improved models is crucial for banks and regulators, and it is
a challenge for statistical physics.
collaborators: Dr. Rudi Schäfer,
Per-Johan Andersson, Andreas Öberg, Markus Sjölin, Andreas
Sundin, Michal Wolanski, Patrik Frisk, Johnny Pégeot, Juan
Manuel Vázquez Montejo, Fredrik Nilsson and Per Berseus
(Examen-students)
external contact: Alexander Koivusalo (Danske Capital, Denmark), Dr. Axel Müller-Groeling (Centrosolar, Hamburg), Eduard Seligman (Osprey Asset Management, Geneva)