## Topological Quantum Matter

**Introduction**. – In the last decade, since the groundbreaking discovery of spin-orbit induced topological insulators, tremendous progress has been made in our understanding of topological materials. However, a complete classification of topological phases in terms of crystal symmetries is still lacking and, in many cases, it is unclear how the topological response functions can be measured in an experiment. One aim of the undergraduate research projects is to study and classify topological magnetic insulators and semi-metals that are protected by crystal symmetries. Another aim is to show that strain-induced pseudo-magnetic fields can be used to probe topological responses. Furthermore, light-induced topological states, i.e., Floquet topological phases, will be investigated. While these projects mainly aim at the design and fundamental understanding of new topological states of matter, the insights gained may have important consequences for applications, such as spintronic, thermoelectric, and superconducting devices, which can be utilized for future information and communication technologies.

**Magnetic insulators and semi-metals with non-symmorphic symmetries.** – Topological semi-metals exhibit band crossings near the Fermi energy, which are protected by the non-trivial topological character of the wave functions. In many cases, these topological band degeneracies give rise to exotic surface states and unusual magneto-transport properties. In this project we will study this physics in the context of magnetic semi-metals with non-symmorphic symmetries. We will start by examining materials with cubic space group symmetries and then extend the analysis to hexagonal systems. We will study and characterize the exotic phenomena associated with the nontrivial topologies of Weyl and Dirac modes that exist in these magnetic topological materials.

**Strain-induced pseudo-magnetic fields.** – Non-uniform strain applied to topological semi-metals induces a pseudo-magnetic field that acts on the low-energy Dirac orWeyl excitations. These pseudo-magnetic fields can be of the order of 300 Tesla, one order of magnitude larger than the strongest man-made conventional magnetic fields. Just like conventional magnetic fields, pseudo-magnetic fields lead to Landau levels in the single particle spectrum. Since the Landau levels have a flat dispersion with a large ground-state degeneracy, they will be unstable in the presence of many-body interactions. In this project we will derive the pseudo-magnetic fields of strained two-dimensional and three-dimensional topological semi-metals. We will investigate, how the pseudo-magnetic fields can be used to measure the topological responses of these semi-metals. We will also study the interaction-induced instabilities of the non-dispersive Landau levels.

**Floquet topological phases.** – Irradiating quantum materials with microwave light leads to a periodic perturbation, thereby adding an extra dimension to the problem. Using Floquet theory, we will study the quasi-energy spectra of periodically driven quantum materials. One aim will be to construct four- and higher-dimensional topological states, such as the four-dimensional Chern insulator, which is characterized by the second Chern number. We will compute the surface states of these topological phases and find new ways how to probe the edges states in the experiment.