The discovery of diverse topological states of matter, including many classes of topological crystalline insulators and topological semimetals, represents the recent development of condensed matter physics. As a theoretical and computational physicist, I would like to share my knowledge that helped me work in this field with those who are not familiar with the concept of topological states. After introducing Laughlin’s argument, I will extend it to discuss a recent technique to evaluate topological indices using the Wilson loop calculation. This will provide a unified perspective to the diverse topological indices characterizing different topological states matter. At the end, I will introduce our recent discovery of the “topological Dirac insulator” [1], which is an exception to symmetry-enhanced fermion doubling theorem. I hope that this talk can be a useful guide to one-hour traveling to the world of topological materials.

[1] “Wallpaper Fermions and the Nonsymmorphic Dirac Insulator”, Benjamin J. Wieder, Barry Bradlyn, Zhijun Wang, Jennifer Cano, Youngkuk Kim, Hyeong-Seok D. Kim, Andrew M. Rappe, C. L. Kane, and B. Andrei Bernevig, Science 361, 246 (2018)