Atoms to topological electronic materials: A bedtime story for beginners

Abstract: In this review, We discussed the theoretical foundation and experimental discovery of different topological electronic states of material in condensed matter. At first, we briefly reviewed the conventional electronic states, which have been realized in band theory of solid. Next, the simplest non-trivial insulating phase (Integer Quantum Hall State) and the concept of topological order in condensed matter electronic system have been introduced. In the following sections, we discussed Quantum Spin Hall (QSH) State in two dimensions (2D), and reviewed the theoretical and experimental developments from 2D QSH state to 3D topological insulators (TI). Subsequently, we gave a brief overview on theoretical and experimental understanding on recently discovered topological Dirac semimetals, Weyl semimetals, three-, six- , and eight-fold degenerate semimetals, and Nodal line semimetals. Then, topological crystalline insulator, which can not be considered as a descendent of Quantum Spin Hall or Integer Quantum Hall insulator, has been introduced. Finally, we discussed the presence of magnetism in some topological materials and its consequence on electronic band structure.