From Higher-Order Topological Superconductors to Topological Quantum Codes.

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The discovery of a new class of materials known as topological insulators has been cause for great excitement over the past decade owing to the possibility of realising novel phenomena such as insulating materials with highly conducting surfaces. The key to these surprising properties lies in the topology of these systems, that is, global, quantised, characteristics of the bulk material that are insensitive to local imperfections such as defects and impurities. The theoretical advancements in describing such materials have allowed for the topological classification of a further class of many-body systems, namely topological superconductors (TSCs).While the classification of TSCs is now fairly established, much current attention has shifted to a new subclass of TSCs known as higher-order TSCs. These systems display even more exotic properties deriving from spatial symmetries. Under certain circumstances, one may predict the existence of exotic corner-bound zero-energy objects that lead to a nonlocally supported ground state degeneracy. Novel examples indicate that, for certain forms of interactions, such systems may also support topologically ordered phases closely related to topological quantum codes. Such codes form one of the most promising schemes for quantum error correction and may lead to quantum computers robust against most relevant sources of disturbances.The goal of this project is to develop a classification of higher-order TSCs and, based on this, to investigate the connection between interacting higher-order TSCs and topological quantum codes. We will build on, and develop further, existing theoretical frameworks for characterising topological materials and correlated forms of topological order. We plan to support our formal theoretical findings with numerical simulations.