# 48th IFF Spring School

Topological Matter - Topological Insulators, Skyrmions and Majoranas

27 March – 07 April 2017 in Jülich, Germany

## Overview

Topology is the branch of mathematics that deals with properties of spaces that are invariant under smooth deformations. It provides newly appreciated mathematical tools in condensed matter physics that are currently revolutionizing the field of quantum matter and materials. Topology dictates that if two different Hamiltonians can be smoothly deformed into each other they give rise to many common physical properties and states are homotopy invariant. Thus, topological invariance, which is often protected by discrete symmetries, provides some robustness that translates into the quantization of properties; such a robust quantization motivates the search and discovery of new topological matter.

So far, the mainstream of modern topological condensed matter physics relies on two profoundly different scenarios: the emergence of the complex topology either in real space, as manifested e.g. in non-trivial magnetic structures or in momentum space, finding its realization in such materials as topological and Chern insulators. The latter renowned class of solids attracted considerable attention in recent years owing to its fascinating properties of spin-momentum locking, emergence of topologically protected surface/edge states governed by Dirac physics, as well as the quantization of Hall conductance and the discovery of the quantum spin Hall effect. Historically, the discovery of topological insulators gave rise to the discovery of a whole plethora of topologically non-trivial materials such as Weyl semimetals or topological superconductors, relevant in the context of the realization of Majorana fermions and topological quantum computation.

At the same time, the physics of skyrmions with complex magnetic real-space topologies is rapidly moving to the centre of attention in spintronics owing to the bright prospects of skyrmionic materials related to topological protection and robust dynamics. The discovery of skyrmions in various geometries (bulk, thin films, interfaces), the complex interplay of their properties with their topology, the fascinating aspects of their dynamics and transport properties are believed to result in skyrmions and other topological spin structures as basic building blocks for information manipulation and storage.

Overall, the expanding manifold of materials, phenomena and concepts, which are born from the combination of ideas and methods of topological characterization and geometrical analysis with the most advanced developments in modern solid state physics, marks one of the most exciting moments in the history of physics related to a paradigm shift in our understanding of matter.