Topological phases and transitions in condensed matter systems

Two months ago, three physicists won the Nobel physics prize for their discovery of topological phases and transitions. In this paper, we review the origin of the concept of topology in condensed matter physics, then present a brief introduction to the main classes of topological states in solid-state materials, including topological insulators, the quantum anomalous Hall effect,topological crystalline insulators, and topological semimetals.