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Three-Dimensional Topological Insulators

Topological insulators in three dimensions are nonmagnetic insulators that possess metallic surface states as a consequence of the nontrivial topology of electronic wavefunctions in the bulk of the material. They are the first known examples of topological order in bulk solids. We review the basic phenomena and experimental history, starting with the observation of topological insulator behavior in Bi$_x$Sb$_{1-x}$ by spin- and angle-resolved photoemission spectroscopy and continuing through measurements on other materials and by other probes. A self-contained introduction to the single-particle theory is then given, followed by the many-particle definition of a topological insulator as a material with quantized magnetoelectric polarizability. The last section reviews recent work on strongly correlated topological insulators and new effects that arise from the proximity effect between a topological insulator and a superconductor. While this article is not intended to be a comprehensive review of what is already a rather large field, we hope that it will serve as a useful introduction, summary of recent progress, and guideline to future directions.