Topological materials: from fundamentals to applications

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***The recent discovery of topological insulators, superconductors and semimetals has revolutionized the understanding of solids by combining quantum physics with topology (a branch of mathematics). In these so--called "topological materials", the electronic structure is characterized by nonzero integers known as "topological invariants". These invariants are immune to the imperfections of the material and manifest themselves physically through unusual electronic states localized at sample boundaries or surfaces. ******The advent of topological materials, a remarkable feat of fundamental science, harbors in addition a significant promise for practical applications. For example, the "Majorana modes" emerging in topological superconductors are promising candidates for quantum computers. In addition, the surface states of topological insulators could help realize low-dissipation magneto-electronic (spintronics) devices.******In spite of the recent advances, the transition of topological materials towards transformative quantum technologies remains in its infancy and faces various challenges. To begin with, Majorana modes are difficult to detect and manipulate because they have no charge and no magnetic properties. Also, potential spintronics devices based on topological materials exploit properties (such as the "spin-momentum locking") that are not unique to topological materials. At this point, it is unclear that these properties are sufficiently efficient for useful devices. ******The objective of our research is to help advance the transition of topological materials from fundamentals to applications by developing novel schemes to detect and manipulate Majorana modes, by introducing new optimization algorithms, and by conceiving spintronics devices that exploit genuine topological effects. Making progress along these directions will not only help the development of useful devices based on topological materials, but will also improve the fundamental understanding of this exciting new class of materials.