In this lecture, numerical tools for quantum many-body physics will be theoretically discussed and practically applied. We will cover mean-field theory, exact diagonalization, tensor network methods such as DMRG, and quantum Monte Carlo methods. The successful participant will be guided with accompanying exercises towards developing own source codes for solving basic quantum many-body problems using the aforementioned methods.

The aim of this course of lectures is an elementary introduction to the foundations of the theory of topological insulators, topological superconductors and topological semimetals. We start with simple one-dimensional topological models, introduce the Berry phase and Chern invariants, and move on to “topological” phases in condensed media. We will discuss how the concept of topology helps us to understand the physical properties of topological materials such as polarization, topologically protected edge states, quantized transverse conductivity, and so on.

1. Introduction 2. Open (quantum) systems 3. Stochastic processes, Kolmogorov Theorem, Markov processes 4. Langevin equation, stochastic differential equations (Ito, Stratonovich) 5. Fokker-Planck equation, (quantum) master equation 6. Path integrals: quantum and classical 7. Stochastic Schrödinger equations 8. Continuous measurement 9. Quantum Markov and non-Markovian processes. To clarify the information university website: this lecture course is in English.

N-body problem; Hamiltonian systems, invariant objects (fixed points, periodic trajectories, invariant tori, and stable and unstable manifolds), nonlinear resonances, bifurcations, Arnold diffusion. Examples: coupled maps, Fermi-Pasta-Ulam-Tsingou problem, spin chains, ...

This class is divided in three parts and will cover hydrodynamics of passive fluids, hydrodynamics of active fluids, and biological pattern formation processes. We will review basics of continuum mechanics and equilibrium and non-equilibrium thermodynamics, linear response theory, Onsager Relations, and the derivation of hydrodynamic equations both from microscopic rules and from a phenomenological approach using broken symmetries and conservation laws. Topics for passive fluids include thin film fluids and the statics and dynamics of nematic liquid crystals. Topics for active fluids include active nematic or polar fluids and biological fluids. Moving beyond linear response we will discuss spatial chemical systems, bifurcation theory, and biological pattern formation. The class will end on coupled chemical and hydrodynamic pattern formation processes relevant for morphogenesis.

The aim of this lecture is to give an insight into the exciting research in the field of magnetism and magnetic materials on the nanoscale. We will start with an introduction in the basics of (ferro)magnetism and magnetic materials with particular focus on magnetic anisotropy, domains and exchange bias and we will give an introduction to magnetic microscopy. Using this knowledge, superparamagnetism and molecular magnets will be discussed. In addition, we will cover aspects of spin transport phenomena such as giant and tunneling magneto resistance, including e.g.a discussion on spin transfer torque.

o Fundamentals of optical spectroscopy o Electromagnetic waves in vacuum and matter o Dielectric function and optical conductivity o Types of material response and their manifestations in optics o Sum rules and Kramers-Kronig relations o Optics of interfaces, surface modes o Screening and Lindhard response function o Modern experimental optical techniques