# Topological quantum systems

Placing many electrons together into a solid state environment often leads to the formation of new and even surprising physical behavior. Under certain conditions, this behavior is aptly described in terms of the underlying topology of the system, most notably in the case of the topological insulators. These relatively new states of electronic matter are characterized by the presence of an energy bandgap within the bulk of the material, while the material’s edge or surface hosts topologically protected gapless modes. In the fractional quantum Hall effect, the nontrivial topology gives rise to fractionally charged elementary excitations which in some cases may even possess non-Abelian braiding statistics. More recently, the discovery of the quantum spin Hall effect has provided another route toward these exotic non-Abelian particles, called Majorana fermions, via superconductivity injected into helical edge states. Our group is currently pursuing the construction of a topological quantum computer via the creation and manipulation of Majorana fermions, using as a host material the quantum well formed by HgTe/HgCdTe heterostructures.