| Abstract: | In the past several years, phases of matter which do not exhibit local order parameters have caused tremendous excitement in condensed matter physics. These topological phases of matter fall outside the Landau-Ginzburg classification of broken symmetry phases, in which a local order parameter such as the magnetization characterizes the state. Topological states of matter now comprise abelian and nonabelian fractional quantum Hall states, spin liquids, and more recently, topological insulators. These are featureless bulk insulating materials, but they exhibit surface states which are perfect metals. In this colloquium, I will describe the discovery both theoretical and experimental of topological insulators, their properties, and the connection to other fields from High-Energy physics to Thermoelectric materials. I will then describe the theory behind the characterization of such insulators, giving intuitive explanations of the topological indices involved. I will present the outstanding problems in the field, including the possible existence of fractional interacting states in three space dimensions. I will close by mentioning recent attempts to classify all topological phases of matter, with emphasis on a new non-local measure, the entanglement between two parts of the system. |
