Mathematical and Theoretical Physics Seminar

Abstract: In disordered Quantum Systems all translational and rotational symmetries are broken and the disorder can only be statistically defined by its distribution function in an ensemble of random systems. Accordingly, all physical properties are fully described only by their distribution functions, such as the distribution of level spacings or wave function intensities. These are determined by the remaining symmetries such as Time Reversal Symmetry, Spin Symmetry, Particle- Hole Symmetry and Parity. This maps the problem of Disordered Quantum Systems on the problem of random Hermitian matrices. Following the classification of symmetric spaces by mathematician Élie Cartan one finds that these can be divided into 10 symmetry classes, which are now all known to be realised in physical systems such as disordered metal particles, quantum chaotic billiards or in the high excitation spectra of nuclei, where all symmetry classes are realized by imposing different discrete symmetries.
Here, we review how we can thereby study disordered quantum systems by the dynamics on a special set of highly symmetric manifolds, the Riemannian symmetric spaces and the corresponding supersymmetric spaces. In particular we focus on the information one can extract by considering averaged autocorrelation functions of spectral determinants (ASD). We review how these are mapped in the functional intergal representation on the problem of the dynamics on compact symmetric spaces which allows us to study noninteracting disordered systems from quantum chaos to Anderson localisation with rigourous methods, albeit with the warning that the obtained results are only valid for the ASD which yields only limited although nontrivial information on the disordered system. We conclude with a review and an outlook to open problems, including the Integer Quantum Hall transition, the Anderson delocalisation transition of noninteracting particles and the theory of disordered interacting many body systems. In particular we argue that the ASD may be a useful guide and yields us useful insights towards the nonperturbative solution of these problems.

Abstract: Systems Biology is a young and rapidly evolving research field, which combines experimental techniques and mathematical modeling in order to achieve a mechanistic understanding of processes underlying the regulation and evolution of living systems. Physics has a long tradition of characterizing and understanding emergent collective behaviors in systems of interacting units and searching for universal laws. Therefore, it is natural that many concepts used in Systems Biology have their roots in Physics. With an emphasis on Theoretical Physics, I will here review the "Physics core" of Systems Biology, show how some success stories in Systems Biology can be traced back to concepts developed in Physics, and discuss how Systems Biology can further benefit from its Theoretical Physics foundation.

Abstract: With new optical clocks and laser interferometers in space new instruments are at work to measure the gravitational field of the Earth with highest precision. The measurement procedures as well as the highest precision make it necessary to employ General Relativity. Accordingly, in the talk a fully general relativistic approach to geodesy is presented. Starting from a stationary situation the fully relativistic geoid is defined. Since relativistic gravity has more degrees of freedom than Newtonian gravity, a second "geoid" has been found which is related to the rotational degrees of freedom of gravity. The methods to practically measure these geoids are described and the deviation from Newtonian gravity is discussed. In the outlook remaining tasks for a formalism for general relativistic geodesy are outlined.

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Abstract: With direct gravitational wave detections as well as the first picture of the shadow of a black hole, there exists now complementary observational evidence that black holes - a priori a theoretical prediction of the best theory of gravity that we have to this day, General Relativity - do, indeed, (seem to) exist in the universe. These recent observations show that the corresponding black holes can be extremely well matched to the Kerr solution, a vacuum solution of the Einstein equation that describes a black hole determined only by its mass and angular momentum. Observable black holes hence seem to be very simple objects that carry no additional structure ("hair"). However, models appearing in theories that try to explain e.g. the nature of dark energy or the inflationary epoch in the primordial universe as well as recent studies within the gauge/gravity duality contain black hole solutions that often carry hair. Moreover, stars made out of scalar fields - so-called "boson stars" - provide viable alternatives to supermassive black holes.
In this talk, I will discuss boson stars and black holes that carry non-trivial scalar hair, respectively. I will motivate the models discussed and explain their applications.

Abstract: The term quasicrystals refers to a special class of materials that have no translational symmetry and yet have high amounts of long-range order—i.e. the Fourier transform of their atomic density consists of discrete delta-peaks. In crystals, this type of long-range order is easily understood to stem from the fact that a finite arrangement repeats itself indefinitely. In quasicrystals, a weaker condition is met—local configurations of any given size everywhere are almost alike at every scale. The discovery of a superconducting quasicrsytal in 2018 begs the question, how do correlated electrons behave in quasicrystals? We found that we could make considerable progress towards an answer to this question by considering the simplest possible model of a superconducting quasicrystal—the one-dimensional Fibonacci hopping model with pairing introduced via the Bogoliubov-de Gennes self-consistent field method. Today, I will primarily share results for proximity-induced superconductivity at a superconductor-quasicrystal interface. The highlight of the talk is the appearance of the topological gap labels of the Fibonacci chain in the amplitude of the induced order parameter.