Phase transition between quantum spin Hall and ordinary insulating phases

We theoretically study the phase transition between the quantum spin Hall (QSH) and insulator phases, which involves a change of the $Z_2$ topological number. We deal with 2D and 3D systems without impurity and interaction. We introduce a parameter $m$ controlling the phase transition, and we study whether the gap closes or not by one-parameter tuning. In general, level repulsion prevents the gap from closing. In fact, the physics of the $Z_2$ topological number is encoded in the problem whether the gap closes by tuning a single parameter. In 2D [1], as well as in the 3D inversion-symmetric systems [2], the gap closes at one point, $m=m_0$, whereas in 3D inversion-asymmetric systems [2], there appears a finite regime for $m$ ($m_1