Hi all! I am currently reading stuff related to quantum hall effect and topological insulators, and have a couple of questions. 1. I read about that band insulators can be classified into two types: topological trivial insulators and topological non-trivial insulators. And there is a topological index related. And I kind of confused with several terms, the Chern number, topological invariants and TKNN number/invariants, stuff like that (I saw these terms over various materials again and again, but I can hardly find one that is simple enough for me to digest.) Are they referring to the same system? And I would like to ask the difference between 2D topological insulator and Quantum hall states as well. 2. I know that topology is involved when we change the Hamiltonian adiabatically, so Berry phase idea follows naturally, but I knid of dun understand why we can consider a gauge transformation in this case? 3. The edge state, how spin-momentum locking possible? This part is kind of important I guess, coz it after all explains why that state is conducting.