Topological states of quantum matter

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Behavior of electrons in solids is governed by the peculiar laws of quantum mechanics. In the quantum world particles often behave as waves, and wave phenomena, such as light, acquire particle-like attributes. Among the triumphs of the 20th century quantum physics is the band theory: it explains, among other things, why certain crystalline solids (i.e. metals) conduct electricity while others are electrical insulators. Band theory also forms the basis for our understanding of semiconductors which in turn underlie most of the technological progress achieved over the past 50 years. Until very recently it was widely believed that the band theory and its applications were fully and completely understood. It came as a great surprise when in 2005 a group of theorists uncovered a completely new aspect of the band theory rooted in the ideas adopted from the area of mathematics known as topology. These pioneering theoretical works and the subsequent experimental studies established the existence of a brand new class of solids known as `topological insulators'. These materials have insulating bulk but, remarkably, their surfaces conduct electricity. This unusual behavior forms a basis for numerous proposed practical applications.