Winding numbers, discriminants and topological phase transitions

The discovery that the band structures of certain insulators are marked by nontrivial topological properties has spawned a wave of publications dealing with such materials. The mathematical framework that underlies many of these publications is often very abstract, and it tends to override the essential physics, thus precluding an intuitive approach to this fascinating topic. Based on known and novel types of model systems, we point out several aspects of topological insulators, which may be approached by elementary and intuitive concepts like the Brouwer degree, winding numbers and the characterization of topological phase transitions using secular equations and discriminant theory.