Topological quantum systems

In the last few decades, novel phases of matter with uniquely quantum character have been discovered that do not fit into the classical symmetry-based classification of phases. These systems, broadly called topological quantum systems or -topologically ordered” systems, often display distinctive features such as long-range entanglement, topological degeneracy, and stability to local perturbations. Topologically ordered systems also allow for exotic quasiparticles called anyonic excitations – particles that exhibit neither bosonic nor fermionic statistics. Topological systems are suspected to arise in natural systems such as fractional quantum hall (FQH) liquids, as well as artificial systems such as those engineered in optical lattices. Their unique properties also give these systems diverse application, most notably to quantum computing. As well as the complicated physics involved in FQH liquids, we can study topological order in much simpler toy models that allow us to see the main phenomena much more directly. The most famous of these models is the toric code invented by Alexei Kitaev. This code is the prototypical example of a simple topologically ordered system, and itself displays the anyonic statistics, long-range order, topological degeneracy and stability that characterize topologically ordered systems. In addition, the toric code has found application in quantum coding theory and helped to invent the field of topological quantum computing. The toric code and related models are also closely related to lattice gauge theories studied in high energy physics, as well as the theory of quantum gravity in (2+1) dimensions. As well as intrinsic topological order, related phenomena also arise in the study of symmetry-protected topological order (SPTO). The most well-known examples of systems with SPTO are topological insulators, which do not have the full topological protection of the toric code or FQH systems, but inherit similar properties in the presence of a symmetry (in the case of topological insulators, time-reversal symmetry). Topological quantum systems also offer connections to such diverse fields as high energy physics, quantum computing, quantum gravity, and of course condensed matter physics. This course assumes familiarity with basic methods of quantum mechanics. Partial course outline: - • Toy model systems with topological order - • Experimental realizations of topological systems - • Mathematical framework of anyon theories and topological quantum field theories - • Generic features of topological quantum systems – topological invariants, topological stability, boundary theories - • Applications of topological systems – quantum memories, quantum information processing - • Symmetry protected topological order & topological insulators Inst. f. Theoretische Physik Modul: Ausgewählte Themen moderner Physik Universität Hannover SoSe 2015 Dozent