Higher order topology in superconducting and interacting electronic systems

Topological states with hinge and corner modes, so-called higher order topological insulators, have been predicted to exist in various crystalline materials such as elementary bismuth and WTe2. I will start with a review of the theoretical foundations of higher-order topology, introducing topological invariants to identify such phases as well as an explanation for their generalized topological bulk-boundary correspondence. Second, I will discuss venues for realizing higher order topology in superconductors, magnetic, and strongly interacting systems. To define (crystalline) topology in superconductors, I will discuss the question how theoretically a trivial reference state has to be defined (an atomic limit), and how a topological classification can be built up on this. For magnetic systems, the possibility of topological Kondo insulators hosting a phase with hinge states upon undergoing a magnetic instability is outlined. Finally, the generalization of the bulk-boundary correspondence of higher-order topology will be elevated to the interacting case, including the possibility of surface topological order.