Questions on Topological Insulators, confused on some terminologies

[chunfaifai]

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.

 

[BigJDubs]

Any answers or references would be helpful! Thanks a lot! </code>1. The Chern number, topological invariants and TKNN number/invariants all refer to the same system, which is the topological insulator. Topological insulators are materials that have an insulating gap in the bulk but allow for conducting states on their edges. Topological insulators can be further divided into two categories: trivial and non-trivial. The difference between a 2D topological insulator and a quantum hall state is that a quantum hall state is an example of a 2D topological insulator. It is also characterized by an integer invariant known as the Hall conductance, which measures the number of conducting channels at the edge of the material. 2. When we consider a gauge transformation in a topological insulator, it is necessary to ensure that the Hamiltonian remains invariant under this transformation. This is done by introducing a Berry phase factor into the transformed Hamiltonian. This Berry phase factor is responsible for maintaining the topology of the system, and thus allows us to consider the adiabatic evolution of the Hamiltonian.3. Spin-momentum locking occurs in topological insulators due to the Berry phase factor mentioned above. This Berry phase factor causes the momentum of the electrons to be correlated with their spin, resulting in the spin-momentum locking phenomenon. This spin-momentum locking is responsible for the conducting nature of the edge states in topological insulators.