4024081 – Topology in Condensed Matter Physics
From elementary quantum mechanics lectures, we know that different states can be distinguished by their quantum numbers, such as momentum, angular momentum, etc. The appearance of these quantum numbers is closely related to symmetry-related invariance under transformations, e.g., translations or rotations. The introduction of concepts of topology into physics makes it possible to identify further, so-called "topological" quantum numbers. Topological aspects have long been known in physics, e.g., from the Dirac hypothesis of the existence of magnetic monopoles (which would explain the quantization of the electric charge), as well as from nuclear physics of the 50s ("Skyrmions"). The enormous variety of topological effects and their fundamental importance in condensed-matter physics has only become apparent in recent times. Today, an outstanding precision of the integer quantum Hall effect (QHE) is understood as a consequence of its topological nature. Furthermore, extraordinary properties of graphene and of other novel materials---topological insulators and superconductors, Weyl semimetals, etc.---are also due to their topological nature. Fractional charges and exotic statistics of low-lying excitations in fractional QHE are topologically imposed and stabilized, as is also the case for quantum spin liquids. Realizations of Majorana excitations in topological systems are of great interest, especially in connection with their potential application for topological quantum computing. Modern solid-state physics would be deprived of many of its most fascinating and intrinsic aspects without topological concepts.