Quantum spin Hall phases

We review our recent theoretical works on the the quantum spin Hall effect. First we compare edge states in various 2D systems, and see whether they are robust or fragile against perturbations. Through the comparisons we see the robust nature of edge states in 2D quantum spin Hall phases. We see how it is protected by the Z_2 topological number, and reveal the nature of the Z_2 topological number by studying the phase transition between the quantum spin Hall and insulator phases. We also review our theoretical proposal of the ultrathin bismuth film as a candidate to the 2D quantum spin Hall system.